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Continuous function

From Wikipedia, the free encyclopedia
(Redirected fromContinuous extension)
Mathematical function with no sudden changes
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematics, acontinuous function is afunction such that a small variation of theargument induces a small variation of thevalue of the function. This implies there are no abrupt changes in value, known asdiscontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. Adiscontinuous function is a function that isnot continuous. Until the 19th century, mathematicians largely relied onintuitive notions of continuity and considered only continuous functions. Theepsilon–delta definition of a limit was introduced to formalize the definition of continuity.

Continuity is one of the core concepts ofcalculus andmathematical analysis, where arguments and values of functions arereal andcomplex numbers. The concept has been generalized to functionsbetween metric spaces andbetween topological spaces. The latter are the most general continuous functions, and their definition is the basis oftopology.

A stronger form of continuity isuniform continuity. Inorder theory, especially indomain theory, a related concept of continuity isScott continuity.

As an example, the functionH(t) denoting the height of a growing flower at timet would be considered continuous. In contrast, the functionM(t) denoting the amount of money in a bank account at timet would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

History

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A form of theepsilon–delta definition of continuity was first given byBernard Bolzano in 1817.Augustin-Louis Cauchy defined continuity ofy=f(x){\displaystyle y=f(x)} as follows: an infinitely small incrementα{\displaystyle \alpha } of the independent variablex always produces an infinitely small changef(x+α)−f(x){\displaystyle f(x+\alpha )-f(x)} of the dependent variabley (see e.g.Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (seemicrocontinuity). The formal definition and the distinction between pointwise continuity anduniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,[1]Karl Weierstrass[2] denied continuity of a function at a pointc unless it was defined at and on both sides ofc, butÉdouard Goursat[3] allowed the function to be defined only at and on one side ofc, andCamille Jordan[4] allowed it even if the function was defined only atc. All three of those nonequivalent definitions of pointwise continuity are still in use.[5]Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given byPeter Gustav Lejeune Dirichlet in 1854.[6]

Real functions

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Definition

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The functionf(x)=1x{\displaystyle f(x)={\tfrac {1}{x}}} is continuous on its domain (R∖{0}{\displaystyle \mathbb {R} \setminus \{0\}}), but is discontinuous atx=0,{\displaystyle x=0,} when considered as apartial function defined on the reals.[7]

Areal function that is afunction fromreal numbers to real numbers can be represented by agraph in theCartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbrokencurve whosedomain is the entire real line. A more mathematically rigorous definition is given below.[8]

Continuity of real functions is usually defined in terms oflimits. A functionf with variablex iscontinuous at thereal numberc, if the limit off(x),{\displaystyle f(x),} asx tends toc, is equal tof(c).{\displaystyle f(c).}

There are several different definitions of the (global) continuity of a function, which depend on the nature of itsdomain.

A function is continuous on anopen interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval(−∞,+∞){\displaystyle (-\infty ,+\infty )} (the wholereal line) is often called simply a continuous function; one also says that such a function iscontinuous everywhere. For example, allpolynomial functions are continuous everywhere.

A function is continuous on asemi-open or aclosed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the functionf(x)=x{\displaystyle f(x)={\sqrt {x}}} is continuous on its whole domain, which is the closed interval[0,+∞).{\displaystyle [0,+\infty ).}

Many commonly encountered functions arepartial functions that have a domain formed by all real numbers, except someisolated points. Examples include thereciprocal functionx↦1x{\textstyle x\mapsto {\frac {1}{x}}} and thetangent functionx↦tan⁡x.{\displaystyle x\mapsto \tan x.} When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.

A partial function isdiscontinuous at a point if the point belongs to thetopological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functionsx↦1x{\textstyle x\mapsto {\frac {1}{x}}} andx↦sin⁡(1x){\textstyle x\mapsto \sin({\frac {1}{x}})} are discontinuous at0, and remain discontinuous whichever value is chosen for defining them at0. A point where a function is discontinuous is called adiscontinuity.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Letf:D→R{\textstyle f:D\to \mathbb {R} } be a function whosedomainD{\displaystyle D} is contained inR{\displaystyle \mathbb {R} } of real numbers.

Possibilities forD{\displaystyle D} include:

In the case of an open interval,a{\displaystyle a} andb{\displaystyle b} do not belong toD{\displaystyle D}, and the valuesf(a){\displaystyle f(a)} andf(b){\displaystyle f(b)} are not defined, and if they are, they do not matter for continuity onD{\displaystyle D}.

Definition in terms of limits of functions

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The functionf iscontinuous at some pointc of its domain if thelimit off(x),{\displaystyle f(x),} asx approachesc through the domain off, exists and is equal tof(c).{\displaystyle f(c).}[9] In mathematical notation, this is written aslimx→cf(x)=f(c).{\displaystyle \lim _{x\to c}{f(x)}=f(c).}In detail this means three conditions: first,f has to be defined atc (guaranteed by the requirement thatc is in the domain off). Second, the limit of that equation has to exist. Third, the value of this limit must equalf(c).{\displaystyle f(c).}

(Here, we have assumed that the domain off does not have anyisolated points.)

Definition in terms of neighborhoods

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Aneighborhood of a pointc is a set that contains, at least, all points within some fixed distance ofc. Intuitively, a function is continuous at a pointc if the range off over the neighborhood ofc shrinks to a single pointf(c){\displaystyle f(c)} as the width of the neighborhood aroundc shrinks to zero. More precisely, a functionf is continuous at a pointc of its domain if, for any neighborhoodN1(f(c)){\displaystyle N_{1}(f(c))} there is a neighborhoodN2(c){\displaystyle N_{2}(c)} in its domain such thatf(x)∈N1(f(c)){\displaystyle f(x)\in N_{1}(f(c))} wheneverx∈N2(c).{\displaystyle x\in N_{2}(c).}

As neighborhoods are defined in anytopological space, this definition of a continuous function applies not only for real functions but also when the domain and thecodomain aretopological spaces and is thus the most general definition. It follows that a function is automatically continuous at everyisolated point of its domain. For example, every real-valued function on the integers is continuous.

Definition in terms of limits of sequences

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The sequenceexp(1/n) converges toexp(0) = 1

One can instead require that for anysequence(xn)n∈N{\displaystyle (x_{n})_{n\in \mathbb {N} }} of points in the domain whichconverges toc, the corresponding sequence(f(xn))n∈N{\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }} converges tof(c).{\displaystyle f(c).} In mathematical notation,∀(xn)n∈N⊂D:limn→∞xn=c⇒limn→∞f(xn)=f(c).{\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

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Illustration of theε-δ-definition: atx = 2, any valueδ ≤ 0.5 satisfies the condition of the definition forε = 0.5.

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a functionf:D→R{\displaystyle f:D\to \mathbb {R} } as above and an elementx0{\displaystyle x_{0}} of the domainD{\displaystyle D},f{\displaystyle f} is said to be continuous at the pointx0{\displaystyle x_{0}} when the following holds: For any positive real numberε>0,{\displaystyle \varepsilon >0,} however small, there exists some positive real numberδ>0{\displaystyle \delta >0} such that for allx{\displaystyle x} in the domain off{\displaystyle f} withx0−δ<x<x0+δ,{\displaystyle x_{0}-\delta <x<x_{0}+\delta ,} the value off(x){\displaystyle f(x)} satisfiesf(x0)−ε<f(x)<f(x0)+ε.{\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .}

Alternatively written, continuity off:D→R{\displaystyle f:D\to \mathbb {R} } atx0∈D{\displaystyle x_{0}\in D} means that for everyε>0,{\displaystyle \varepsilon >0,} there exists aδ>0{\displaystyle \delta >0} such that for allx∈D{\displaystyle x\in D}:|x−x0|<δ   implies   |f(x)−f(x0)|<ε.{\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .}

More intuitively, we can say that if we want to get all thef(x){\displaystyle f(x)} values to stay in some smallneighborhood aroundf(x0),{\displaystyle f\left(x_{0}\right),} we need to choose a small enough neighborhood for thex{\displaystyle x} values aroundx0.{\displaystyle x_{0}.} If we can do that no matter how small thef(x0){\displaystyle f(x_{0})} neighborhood is, thenf{\displaystyle f} is continuous atx0.{\displaystyle x_{0}.}

In modern terms, this is generalized by the definition of continuity of a function with respect to abasis for the topology, here themetric topology.

Weierstrass had required that the intervalx0−δ<x<x0+δ{\displaystyle x_{0}-\delta <x<x_{0}+\delta } be entirely within the domainD{\displaystyle D}, but Jordan removed that restriction.

Definition in terms of control of the remainder

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In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A functionC:[0,∞)→[0,∞]{\displaystyle C:[0,\infty )\to [0,\infty ]} is called a control function if

A functionf:D→R{\displaystyle f:D\to R} isC-continuous atx0{\displaystyle x_{0}} if there exists such a neighbourhoodN(x0){\textstyle N(x_{0})} that|f(x)−f(x0)|≤C(|x−x0|) for all x∈D∩N(x0){\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})}

A function is continuous inx0{\displaystyle x_{0}} if it isC-continuous for some control functionC.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functionsC{\displaystyle {\mathcal {C}}} a function isC{\displaystyle {\mathcal {C}}}-continuous if it isC{\displaystyle C}-continuous for someC∈C.{\displaystyle C\in {\mathcal {C}}.} For example, theLipschitz, theHölder continuous functions of exponentα and theuniformly continuous functions below are defined by the set of control functionsCLipschitz={C:C(δ)=K|δ|, K>0}{\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}}CHölder−α={C:C(δ)=K|δ|α, K>0}{\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}}Cuniform cont.={C:C(0)=0}{\displaystyle {\mathcal {C}}_{\text{uniform cont.}}=\{C:C(0)=0\}}respectively.

Definition using oscillation

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The failure of a function to be continuous at a point is quantified by itsoscillation.

Continuity can also be defined in terms ofoscillation: a functionf is continuous at a pointx0{\displaystyle x_{0}} if and only if its oscillation at that point is zero;[10] in symbols,ωf(x0)=0.{\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that itquantifies discontinuity: the oscillation gives howmuch the function is discontinuous at a point.

This definition is helpful indescriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less thanε{\displaystyle \varepsilon } (hence aGδ{\displaystyle G_{\delta }} set) – and gives a rapid proof of one direction of theLebesgue integrability condition.[11]

The oscillation is equivalent to theε−δ{\displaystyle \varepsilon -\delta } definition by a simple re-arrangement and by using a limit (lim sup,lim inf) to define oscillation: if (at a given point) for a givenε0{\displaystyle \varepsilon _{0}} there is noδ{\displaystyle \delta } that satisfies theε−δ{\displaystyle \varepsilon -\delta } definition, then the oscillation is at leastε0,{\displaystyle \varepsilon _{0},} and conversely if for everyε{\displaystyle \varepsilon } there is a desiredδ,{\displaystyle \delta ,} the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to ametric space.

Definition using the hyperreals

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Cauchy defined the continuity of a function in the following intuitive terms: aninfinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (seeCours d'analyse, page 34).Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form thehyperreal numbers. In nonstandard analysis, continuity can be defined as follows.

A real-valued functionf is continuous atx if its natural extension to the hyperreals has the property that for all infinitesimaldx,f(x+dx)−f(x){\displaystyle f(x+dx)-f(x)} is infinitesimal[12]

(seemicrocontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression toAugustin-Louis Cauchy's definition of continuity.

Rules for continuity

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The graph of acubic function has no jumps or holes. The function is continuous.

Proving the continuity of a function by a direct application of the definition is generaly a noneasy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:

The graph of a continuousrational function. The function is not defined forx=−2.{\displaystyle x=-2.} The vertical and horizontal lines areasymptotes.

These rules imply that everypolynomial function is continuous everywhere and that arational function is continuous everywhere where it is defined, if the numerator and the denominator have no commonzeros. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

The sinc and the cos functions

An example of a function for which the above rules are not sufficirent is thesinc function, which is defined by⁠sinc⁡(0)=1{\displaystyle \operatorname {sinc} (0)=1}⁠ and⁠sinc⁡(x)=sin⁡xx{\displaystyle \operatorname {sinc} (x)={\tfrac {\sin x}{x}}}⁠ for⁠x≠0{\displaystyle x\neq 0}⁠. The above rules show immediately that the function is continuous for⁠x≠0{\displaystyle x\neq 0}⁠, but, for proving the continuity at⁠0{\displaystyle 0}⁠, one has to provelimx→0sin⁡xx=1.{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1.}As this is true, one gets that the sinc function is continuous function on all real numbers.

Examples of discontinuous functions

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Plot of the signum function. It shows thatlimn→∞sgn⁡(1n)≠sgn⁡(limn→∞1n){\displaystyle \lim _{n\to \infty }\operatorname {sgn} \left({\tfrac {1}{n}}\right)\neq \operatorname {sgn} \left(\lim _{n\to \infty }{\tfrac {1}{n}}\right)}. Thus, the signum function is discontinuous at 0 (seesection 2.1.3).

An example of a discontinuous function is theHeaviside step functionH{\displaystyle H}, defined byH(x)={1 if x≥00 if x<0{\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}}

Pick for instanceε=1/2{\displaystyle \varepsilon =1/2}. Then there is noδ{\displaystyle \delta }-neighborhood aroundx=0{\displaystyle x=0}, i.e. no open interval(−δ,δ){\displaystyle (-\delta ,\;\delta )} withδ>0,{\displaystyle \delta >0,} that will force all theH(x){\displaystyle H(x)} values to be within theε{\displaystyle \varepsilon }-neighborhood ofH(0){\displaystyle H(0)}, i.e. within(1/2,3/2){\displaystyle (1/2,\;3/2)}. Intuitively, we can think of this type of discontinuity as a suddenjump in function values.

Similarly, thesignum or sign functionsgn⁡(x)={ 1 if x>0 0 if x=0−1 if x<0{\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}is discontinuous atx=0{\displaystyle x=0} but continuous everywhere else. Yet another example: the functionf(x)={sin⁡(x−2) if x≠00 if x=0{\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}is continuous everywhere apart fromx=0{\displaystyle x=0}.

Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.

Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coinedpathological, for example,Thomae's function,f(x)={1 if x=01q if x=pq(in lowest terms) is a rational number0 if x is irrational.{\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}}is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein,Dirichlet's function, theindicator function for the set of rational numbers,D(x)={0 if x is irrational (∈R∖Q)1 if x is rational (∈Q){\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{ is irrational }}(\in \mathbb {R} \setminus \mathbb {Q} )\\1&{\text{ if }}x{\text{ is rational }}(\in \mathbb {Q} )\end{cases}}}is nowhere continuous.

Properties

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A useful lemma

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Letf(x){\displaystyle f(x)} be a function that is continuous at a pointx0,{\displaystyle x_{0},} andy0{\displaystyle y_{0}} be a value suchf(x0)≠y0.{\displaystyle f\left(x_{0}\right)\neq y_{0}.} Thenf(x)≠y0{\displaystyle f(x)\neq y_{0}} throughout some neighbourhood ofx0.{\displaystyle x_{0}.}[13]

Proof: By the definition of continuity, takeε=|y0−f(x0)|2>0{\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0} , then there existsδ>0{\displaystyle \delta >0} such that|f(x)−f(x0)|<|y0−f(x0)|2 whenever |x−x0|<δ{\displaystyle \left|f(x)-f(x_{0})\right|<{\frac {\left|y_{0}-f(x_{0})\right|}{2}}\quad {\text{ whenever }}\quad |x-x_{0}|<\delta }Suppose there is a point in the neighbourhood|x−x0|<δ{\displaystyle |x-x_{0}|<\delta } for whichf(x)=y0;{\displaystyle f(x)=y_{0};} then we have the contradiction|f(x0)−y0|<|f(x0)−y0|2.{\displaystyle \left|f(x_{0})-y_{0}\right|<{\frac {\left|f(x_{0})-y_{0}\right|}{2}}.}

Intermediate value theorem

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Theintermediate value theorem is anexistence theorem, based on the real number property ofcompleteness, and states:

If the real-valued functionf is continuous on theclosed interval[a,b],{\displaystyle [a,b],} andk is some number betweenf(a){\displaystyle f(a)} andf(b),{\displaystyle f(b),} then there is some numberc∈[a,b],{\displaystyle c\in [a,b],} such thatf(c)=k.{\displaystyle f(c)=k.}

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, iff is continuous on[a,b]{\displaystyle [a,b]} andf(a){\displaystyle f(a)} andf(b){\displaystyle f(b)} differ insign, then, at some pointc∈[a,b],{\displaystyle c\in [a,b],}f(c){\displaystyle f(c)} must equalzero.

Extreme value theorem

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Theextreme value theorem states that if a functionf is defined on a closed interval[a,b]{\displaystyle [a,b]} (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there existsc∈[a,b]{\displaystyle c\in [a,b]} withf(c)≥f(x){\displaystyle f(c)\geq f(x)} for allx∈[a,b].{\displaystyle x\in [a,b].} The same is true of the minimum off. These statements are not, in general, true if the function is defined on an open interval(a,b){\displaystyle (a,b)} (or any set that is not both closed and bounded), as, for example, the continuous functionf(x)=1x,{\displaystyle f(x)={\frac {1}{x}},} defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Relation to differentiability and integrability

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Everydifferentiable functionf:(a,b)→R{\displaystyle f:(a,b)\to \mathbb {R} }is continuous, as can be shown. Theconverse does not hold: for example, theabsolute value function

f(x)=|x|={ x if x≥0−x if x<0{\displaystyle f(x)=|x|={\begin{cases}\;\;\ x&{\text{ if }}x\geq 0\\-x&{\text{ if }}x<0\end{cases}}}

is everywhere continuous. However, it is not differentiable atx=0{\displaystyle x=0} (but is so everywhere else).Weierstrass's function is also everywhere continuous but nowhere differentiable.

Thederivativef′(x) of a differentiable functionf(x) need not be continuous. Iff′(x) is continuous,f(x) is said to becontinuously differentiable. The set of such functions is denotedC1((a,b)).{\displaystyle C^{1}((a,b)).} More generally, the set of functionsf:Ω→R{\displaystyle f:\Omega \to \mathbb {R} }(from an open interval (oropen subset ofR{\displaystyle \mathbb {R} })Ω{\displaystyle \Omega } to the reals) such thatf isn{\displaystyle n} times differentiable and such that then{\displaystyle n}-th derivative off is continuous is denotedCn(Ω).{\displaystyle C^{n}(\Omega ).} Seedifferentiability class. In the field of computer graphics, properties related (but not identical) toC0,C1,C2{\displaystyle C^{0},C^{1},C^{2}} are sometimes calledG0{\displaystyle G^{0}} (continuity of position),G1{\displaystyle G^{1}} (continuity of tangency), andG2{\displaystyle G^{2}} (continuity of curvature); seeSmoothness of curves and surfaces.

Every continuous functionf:[a,b]→R{\displaystyle f:[a,b]\to \mathbb {R} }isintegrable (for example in the sense of theRiemann integral). The converse does not hold, as the (integrable but discontinuous)sign function shows.

Pointwise and uniform limits

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A sequence of continuous functionsfn(x){\displaystyle f_{n}(x)} whose (pointwise) limit functionf(x){\displaystyle f(x)} is discontinuous. The convergence is not uniform.

Given asequencef1,f2,…:I→R{\displaystyle f_{1},f_{2},\dotsc :I\to \mathbb {R} }of functions such that the limitf(x):=limn→∞fn(x){\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)}exists for allx∈D,{\displaystyle x\in D,}, the resulting functionf(x){\displaystyle f(x)} is referred to as thepointwise limit of the sequence of functions(fn)n∈N.{\displaystyle \left(f_{n}\right)_{n\in N}.} The pointwise limit function need not be continuous, even if all functionsfn{\displaystyle f_{n}} are continuous, as the animation at the right shows. However,f is continuous if all functionsfn{\displaystyle f_{n}} are continuous and the sequenceconverges uniformly, by theuniform convergence theorem. This theorem can be used to show that theexponential functions,logarithms,square root function, andtrigonometric functions are continuous.

Directional Continuity

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  • A right-continuous function
    A right-continuous function
  • A left-continuous function
    A left-continuous function

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) andsemi-continuity. Roughly speaking, a function isright-continuous if no jump occurs when the limit point is approached from the right. Formally,f is said to be right-continuous at the pointc if the following holds: For any numberε>0{\displaystyle \varepsilon >0} however small, there exists some numberδ>0{\displaystyle \delta >0} such that for allx in the domain withc<x<c+δ,{\displaystyle c<x<c+\delta ,} the value off(x){\displaystyle f(x)} will satisfy|f(x)−f(c)|<ε.{\displaystyle |f(x)-f(c)|<\varepsilon .}

This is the same condition as continuous functions, except it is required to hold forx strictly larger thanc only. Requiring it instead for allx withc−δ<x<c{\displaystyle c-\delta <x<c} yields the notion ofleft-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.

Semicontinuity

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Main article:Semicontinuity

A functionf islower semi-continuous if, roughly, any jumps that might occur only go down, but not up. That is, for anyε>0,{\displaystyle \varepsilon >0,} there exists some numberδ>0{\displaystyle \delta >0} such that for allx in the domain with|x−c|<δ,{\displaystyle |x-c|<\delta ,} the value off(x){\displaystyle f(x)} satisfiesf(x)≥f(c)−ϵ.{\displaystyle f(x)\geq f(c)-\epsilon .}The reverse condition isupper semi-continuity.

Continuous functions between metric spaces

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The concept of continuous real-valued functions can be generalized to functions betweenmetric spaces. A metric space is a setX{\displaystyle X} equipped with a function (calledmetric)dX,{\displaystyle d_{X},} that can be thought of as a measurement of the distance of any two elements inX. Formally, the metric is a functiondX:X×X→R{\displaystyle d_{X}:X\times X\to \mathbb {R} }that satisfies a number of requirements, notably thetriangle inequality. Given two metric spaces(X,dX){\displaystyle \left(X,d_{X}\right)} and(Y,dY){\displaystyle \left(Y,d_{Y}\right)} and a functionf:X→Y{\displaystyle f:X\to Y}thenf{\displaystyle f} is continuous at the pointc∈X{\displaystyle c\in X} (with respect to the given metrics) if for any positive real numberε>0,{\displaystyle \varepsilon >0,} there exists a positive real numberδ>0{\displaystyle \delta >0} such that allx∈X{\displaystyle x\in X} satisfyingdX(x,c)<δ{\displaystyle d_{X}(x,c)<\delta } will also satisfydY(f(x),f(c))<ε.{\displaystyle d_{Y}(f(x),f(c))<\varepsilon .} As in the case of real functions above, this is equivalent to the condition that for every sequence(xn){\displaystyle \left(x_{n}\right)} inX{\displaystyle X} with limitlimxn=c,{\displaystyle \lim x_{n}=c,} we havelimf(xn)=f(c).{\displaystyle \lim f\left(x_{n}\right)=f(c).} The latter condition can be weakened as follows:f{\displaystyle f} is continuous at the pointc{\displaystyle c} if and only if for every convergent sequence(xn){\displaystyle \left(x_{n}\right)} inX{\displaystyle X} with limitc{\displaystyle c}, the sequence(f(xn)){\displaystyle \left(f\left(x_{n}\right)\right)} is aCauchy sequence, andc{\displaystyle c} is in the domain off{\displaystyle f}.

The set of points at which a function between metric spaces is continuous is aGδ{\displaystyle G_{\delta }} set – this follows from theε−δ{\displaystyle \varepsilon -\delta } definition of continuity.

This notion of continuity is applied, for example, infunctional analysis. A key statement in this area says that alinear operatorT:V→W{\displaystyle T:V\to W}betweennormed vector spacesV{\displaystyle V} andW{\displaystyle W} (which arevector spaces equipped with a compatiblenorm, denoted‖x‖{\displaystyle \|x\|}) is continuous if and only if it isbounded, that is, there is a constantK{\displaystyle K} such that‖T(x)‖≤K‖x‖{\displaystyle \|T(x)\|\leq K\|x\|}for allx∈V.{\displaystyle x\in V.}

Uniform, Hölder and Lipschitz continuity

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For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone.

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the wayδ{\displaystyle \delta } depends onε{\displaystyle \varepsilon } andc in the definition above. Intuitively, a functionf as above isuniformly continuous if theδ{\displaystyle \delta } doesnot depend on the pointc. More precisely, it is required that for everyreal numberε>0{\displaystyle \varepsilon >0} there existsδ>0{\displaystyle \delta >0} such that for everyc,b∈X{\displaystyle c,b\in X} withdX(b,c)<δ,{\displaystyle d_{X}(b,c)<\delta ,} we have thatdY(f(b),f(c))<ε.{\displaystyle d_{Y}(f(b),f(c))<\varepsilon .} Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain spaceX iscompact. Uniformly continuous maps can be defined in the more general situation ofuniform spaces.[14]

A function isHölder continuous with exponent α (a real number) if there is a constantK such that for allb,c∈X,{\displaystyle b,c\in X,} the inequalitydY(f(b),f(c))≤K⋅(dX(b,c))α{\displaystyle d_{Y}(f(b),f(c))\leq K\cdot (d_{X}(b,c))^{\alpha }}holds. Any Hölder continuous function is uniformly continuous. The particular caseα=1{\displaystyle \alpha =1} is referred to asLipschitz continuity. That is, a function is Lipschitz continuous if there is a constantK such that the inequalitydY(f(b),f(c))≤K⋅dX(b,c){\displaystyle d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)}holds for anyb,c∈X.{\displaystyle b,c\in X.}[15] The Lipschitz condition occurs, for example, in thePicard–Lindelöf theorem concerning the solutions ofordinary differential equations.

Continuous functions between topological spaces

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Another, more abstract, notion of continuity is the continuity of functions betweentopological spaces in which there generally is no formal notion of distance, as there is in the case ofmetric spaces. A topological space is a setX together with a topology onX, which is a set ofsubsets ofX satisfying a few requirements with respect to their unions and intersections that generalize the properties of theopen balls in metric spaces while still allowing one to talk about the neighborhoods of a given point. The elements of a topology are calledopen subsets ofX (with respect to the topology).

A functionf:X→Y{\displaystyle f:X\to Y}between two topological spacesX andY is continuous if for every open setV⊆Y,{\displaystyle V\subseteq Y,} theinverse imagef−1(V)={x∈X|f(x)∈V}{\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}}is an open subset ofX. That is,f is a function between the setsX andY (not on the elements of the topologyTX{\displaystyle T_{X}}), but the continuity off depends on the topologies used onX andY.

This is equivalent to the condition that thepreimages of theclosed sets (which are the complements of the open subsets) inY are closed inX.

An extreme example: if a setX is given thediscrete topology (in which every subset is open), all functionsf:X→T{\displaystyle f:X\to T}to any topological spaceT are continuous. On the other hand, ifX is equipped with theindiscrete topology (in which the only open subsets are the empty set andX) and the spaceT set is at leastT0, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

Continuity at a point

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Continuity at a point: For every neighborhoodV off(x){\displaystyle f(x)}, there is a neighborhoodU ofx such thatf(U)⊆V{\displaystyle f(U)\subseteq V}

The translation in the language of neighborhoods of the(ε,δ){\displaystyle (\varepsilon ,\delta )}-definition of continuity leads to the following definition of the continuity at a point:

A functionf:X→Y{\displaystyle f:X\to Y} is continuous at a pointx∈X{\displaystyle x\in X} if and only if for any neighborhoodV off(x){\displaystyle f(x)} inY, there is a neighborhoodU ofx{\displaystyle x} such thatf(U)⊆V.{\displaystyle f(U)\subseteq V.}

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by usingpreimages rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, andf−1(V){\displaystyle f^{-1}(V)} is the largest subsetU ofX such thatf(U)⊆V,{\displaystyle f(U)\subseteq V,} this definition may be simplified into:

A functionf:X→Y{\displaystyle f:X\to Y} is continuous at a pointx∈X{\displaystyle x\in X} if and only iff−1(V){\displaystyle f^{-1}(V)} is a neighborhood ofx{\displaystyle x} for every neighborhoodV off(x){\displaystyle f(x)} inY.

As an open set is a set that is a neighborhood of all its points, a functionf:X→Y{\displaystyle f:X\to Y} is continuous at every point ofX if and only if it is a continuous function.

IfX andY are metric spaces, it is equivalent to consider theneighborhood system ofopen balls centered atx andf(x) instead of all neighborhoods. This gives back the aboveε−δ{\displaystyle \varepsilon -\delta } definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is aHausdorff space, it is still true thatf is continuous ata if and only if the limit off asx approachesa isf(a). At an isolated point, every function is continuous.

Givenx∈X,{\displaystyle x\in X,} a mapf:X→Y{\displaystyle f:X\to Y} is continuous atx{\displaystyle x} if and only if wheneverB{\displaystyle {\mathcal {B}}} is a filter onX{\displaystyle X} thatconverges tox{\displaystyle x} inX,{\displaystyle X,} which is expressed by writingB→x,{\displaystyle {\mathcal {B}}\to x,} then necessarilyf(B)→f(x){\displaystyle f({\mathcal {B}})\to f(x)} inY.{\displaystyle Y.} IfN(x){\displaystyle {\mathcal {N}}(x)} denotes theneighborhood filter atx{\displaystyle x} thenf:X→Y{\displaystyle f:X\to Y} is continuous atx{\displaystyle x} if and only iff(N(x))→f(x){\displaystyle f({\mathcal {N}}(x))\to f(x)} inY.{\displaystyle Y.}[16] Moreover, this happens if and only if theprefilterf(N(x)){\displaystyle f({\mathcal {N}}(x))} is afilter base for the neighborhood filter off(x){\displaystyle f(x)} inY.{\displaystyle Y.}[16]

Alternative definitions

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Severalequivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.

Sequences and nets

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In several contexts, the topology of a space is conveniently specified in terms oflimit points. This is often accomplished by specifying when a point is thelimit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of pointsindexed by adirected set, known asnets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a functionf:X→Y{\displaystyle f:X\to Y} issequentially continuous if whenever a sequence(xn){\displaystyle \left(x_{n}\right)} inX{\displaystyle X} converges to a limitx,{\displaystyle x,} the sequence(f(xn)){\displaystyle \left(f\left(x_{n}\right)\right)} converges tof(x).{\displaystyle f(x).} Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. IfX{\displaystyle X} is afirst-countable space andcountable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, ifX{\displaystyle X} is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are calledsequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:[17]

Theorem—A functionf:A⊆R→R{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} } is continuous atx0{\displaystyle x_{0}} if and only if it issequentially continuous at that point.

Proof

Proof. Assume thatf:A⊆R→R{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} } is continuous atx0{\displaystyle x_{0}} (in the sense ofϵ−δ{\displaystyle \epsilon -\delta } continuity). Let(xn)n≥1{\displaystyle \left(x_{n}\right)_{n\geq 1}} be a sequence converging atx0{\displaystyle x_{0}} (such a sequence always exists, for example,xn=x, for all n{\displaystyle x_{n}=x,{\text{ for all }}n}); sincef{\displaystyle f} is continuous atx0{\displaystyle x_{0}}∀ϵ>0∃δϵ>0:0<|x−x0|<δϵ⟹|f(x)−f(x0)|<ϵ.(∗){\displaystyle \forall \epsilon >0\,\exists \delta _{\epsilon }>0:0<|x-x_{0}|<\delta _{\epsilon }\implies |f(x)-f(x_{0})|<\epsilon .\quad (*)}For any suchδϵ{\displaystyle \delta _{\epsilon }} we can find a natural numberνϵ>0{\displaystyle \nu _{\epsilon }>0} such that for alln>νϵ,{\displaystyle n>\nu _{\epsilon },}|xn−x0|<δϵ,{\displaystyle |x_{n}-x_{0}|<\delta _{\epsilon },}since(xn){\displaystyle \left(x_{n}\right)} converges atx0{\displaystyle x_{0}}; combining this with(∗){\displaystyle (*)} we obtain∀ϵ>0∃νϵ>0:∀n>νϵ|f(xn)−f(x0)|<ϵ.{\displaystyle \forall \epsilon >0\,\exists \nu _{\epsilon }>0:\forall n>\nu _{\epsilon }\quad |f(x_{n})-f(x_{0})|<\epsilon .}Assume on the contrary thatf{\displaystyle f} is sequentially continuous and proceed by contradiction: supposef{\displaystyle f} is not continuous atx0{\displaystyle x_{0}}∃ϵ>0:∀δϵ>0,∃xδϵ:0<|xδϵ−x0|<δϵ⟹|f(xδϵ)−f(x0)|>ϵ{\displaystyle \exists \epsilon >0:\forall \delta _{\epsilon }>0,\,\exists x_{\delta _{\epsilon }}:0<|x_{\delta _{\epsilon }}-x_{0}|<\delta _{\epsilon }\implies |f(x_{\delta _{\epsilon }})-f(x_{0})|>\epsilon }then we can takeδϵ=1/n,∀n>0{\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0} and call the corresponding pointxδϵ=:xn{\displaystyle x_{\delta _{\epsilon }}=:x_{n}}: in this way we have defined a sequence(xn)n≥1{\displaystyle (x_{n})_{n\geq 1}} such that∀n>0|xn−x0|<1n,|f(xn)−f(x0)|>ϵ{\displaystyle \forall n>0\quad |x_{n}-x_{0}|<{\frac {1}{n}},\quad |f(x_{n})-f(x_{0})|>\epsilon }by constructionxn→x0{\displaystyle x_{n}\to x_{0}} butf(xn)↛f(x0){\displaystyle f(x_{n})\not \to f(x_{0})}, which contradicts the hypothesis of sequential continuity.◼{\displaystyle \blacksquare }

Closure operator and interior operator definitions

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In terms of theinterior andclosure operators, we have the following equivalences,

Theorem—Letf:X→Y{\displaystyle f:X\to Y} be a mapping between topological spaces. Then the following are equivalent.

  1. f{\displaystyle f} is continuous;
  2. for every subsetB⊆Y,{\displaystyle B\subseteq Y,}f−1(intY⁡B)⊆intX⁡(f−1(B));{\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right);}
  3. for every subsetA⊆X,{\displaystyle A\subseteq X,}f(clX⁡A)⊆clY⁡(f(A)).{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}
Proof

Proof. i ⇒ ii. Fix a subsetB{\displaystyle B} ofY.{\displaystyle Y.} SinceintY⁡B{\displaystyle \operatorname {int} _{Y}B} is open.andf{\displaystyle f} is continuous,f−1(intY⁡B){\displaystyle f^{-1}(\operatorname {int} _{Y}B)} is open inX.{\displaystyle X.}AsintY⁡B⊆B,{\displaystyle \operatorname {int} _{Y}B\subseteq B,} we havef−1(intY⁡B)⊆f−1(B).{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq f^{-1}(B).}By the definition of the interior,intX⁡(f−1(B)){\displaystyle \operatorname {int} _{X}\left(f^{-1}(B)\right)} is the largest open set contained inf−1(B).{\displaystyle f^{-1}(B).} Hencef−1(intY⁡B)⊆intX⁡(f−1(B)).{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right).}

ii ⇒ iii. FixA⊆X{\displaystyle A\subseteq X} and letx∈clX⁡A.{\displaystyle x\in \operatorname {cl} _{X}A.} Suppose to the contrary thatf(x)∉clY⁡(f(A)),{\displaystyle f(x)\notin \operatorname {cl} _{Y}\left(f(A)\right),}then we may find some open neighbourhoodV{\displaystyle V} off(x){\displaystyle f(x)} that is disjoint fromclY⁡(f(A)){\displaystyle \operatorname {cl} _{Y}\left(f(A)\right)}. Byii,f−1(V)=f−1(intY⁡V)⊆intX⁡(f−1(V)),{\displaystyle f^{-1}(V)=f^{-1}(\operatorname {int} _{Y}V)\subseteq \operatorname {int} _{X}\left(f^{-1}(V)\right),} hencef−1(V){\displaystyle f^{-1}(V)} is open. Then we have found an open neighbourhood ofx{\displaystyle x} that does not intersectclX⁡A{\displaystyle \operatorname {cl} _{X}A}, contradicting the fact thatx∈clX⁡A.{\displaystyle x\in \operatorname {cl} _{X}A.}Hencef(clX⁡A)⊆clY⁡(f(A)).{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}

iii ⇒ i. LetN⊆Y{\displaystyle N\subseteq Y} be closed. LetM=f−1(N){\displaystyle M=f^{-1}(N)} be the preimage ofN.{\displaystyle N.}Byiii, we havef(clX⁡M)⊆clY⁡(f(M)).{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}\left(f(M)\right).}Sincef(M)=f(f−1(N))⊆N,{\displaystyle f(M)=f(f^{-1}(N))\subseteq N,}we have further thatf(clX⁡M)⊆clY⁡N=N.{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}N=N.}ThusclX⁡M⊆f−1(f(clX⁡M))⊆f−1(N)=M.{\displaystyle \operatorname {cl} _{X}M\subseteq f^{-1}\left(f(\operatorname {cl} _{X}M)\right)\subseteq f^{-1}(N)=M.}HenceM{\displaystyle M} is closed and we are done.

If we declare that a pointx{\displaystyle x} isclose to a subsetA⊆X{\displaystyle A\subseteq X} ifx∈clX⁡A,{\displaystyle x\in \operatorname {cl} _{X}A,} then this terminology allows for aplain English description of continuity:f{\displaystyle f} is continuous if and only if for every subsetA⊆X,{\displaystyle A\subseteq X,}f{\displaystyle f} maps points that are close toA{\displaystyle A} to points that are close tof(A).{\displaystyle f(A).} Similarly,f{\displaystyle f} is continuous at a fixed given pointx∈X{\displaystyle x\in X} if and only if wheneverx{\displaystyle x} is close to a subsetA⊆X,{\displaystyle A\subseteq X,} thenf(x){\displaystyle f(x)} is close tof(A).{\displaystyle f(A).}

Instead of specifying topological spaces by theiropen subsets, any topology onX{\displaystyle X} canalternatively be determined by aclosure operator or by aninterior operator. Specifically, the map that sends a subsetA{\displaystyle A} of a topological spaceX{\displaystyle X} to itstopological closureclX⁡A{\displaystyle \operatorname {cl} _{X}A} satisfies theKuratowski closure axioms. Conversely, for anyclosure operatorA↦cl⁡A{\displaystyle A\mapsto \operatorname {cl} A} there exists a unique topologyτ{\displaystyle \tau } onX{\displaystyle X} (specifically,τ:={X∖cl⁡A:A⊆X}{\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}) such that for every subsetA⊆X,{\displaystyle A\subseteq X,}cl⁡A{\displaystyle \operatorname {cl} A} is equal to the topological closurecl(X,τ)⁡A{\displaystyle \operatorname {cl} _{(X,\tau )}A} ofA{\displaystyle A} in(X,τ).{\displaystyle (X,\tau ).} If the setsX{\displaystyle X} andY{\displaystyle Y} are each associated with closure operators (both denoted bycl{\displaystyle \operatorname {cl} }) then a mapf:X→Y{\displaystyle f:X\to Y} is continuous if and only iff(cl⁡A)⊆cl⁡(f(A)){\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))} for every subsetA⊆X.{\displaystyle A\subseteq X.}

Similarly, the map that sends a subsetA{\displaystyle A} ofX{\displaystyle X} to itstopological interiorintX⁡A{\displaystyle \operatorname {int} _{X}A} defines aninterior operator. Conversely, any interior operatorA↦int⁡A{\displaystyle A\mapsto \operatorname {int} A} induces a unique topologyτ{\displaystyle \tau } onX{\displaystyle X} (specifically,τ:={int⁡A:A⊆X}{\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}) such that for everyA⊆X,{\displaystyle A\subseteq X,}int⁡A{\displaystyle \operatorname {int} A} is equal to the topological interiorint(X,τ)⁡A{\displaystyle \operatorname {int} _{(X,\tau )}A} ofA{\displaystyle A} in(X,τ).{\displaystyle (X,\tau ).} If the setsX{\displaystyle X} andY{\displaystyle Y} are each associated with interior operators (both denoted byint{\displaystyle \operatorname {int} }) then a mapf:X→Y{\displaystyle f:X\to Y} is continuous if and only iff−1(int⁡B)⊆int⁡(f−1(B)){\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)} for every subsetB⊆Y.{\displaystyle B\subseteq Y.}[18]

Filters and prefilters

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Main article:Filters in topology

Continuity can also be characterized in terms offilters. A functionf:X→Y{\displaystyle f:X\to Y} is continuous if and only if whenever a filterB{\displaystyle {\mathcal {B}}} onX{\displaystyle X}converges inX{\displaystyle X} to a pointx∈X,{\displaystyle x\in X,} then theprefilterf(B){\displaystyle f({\mathcal {B}})} converges inY{\displaystyle Y} tof(x).{\displaystyle f(x).} This characterization remains true if the word "filter" is replaced by "prefilter."[16]

Properties

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Iff:X→Y{\displaystyle f:X\to Y} andg:Y→Z{\displaystyle g:Y\to Z} are continuous, then so is the compositiong∘f:X→Z.{\displaystyle g\circ f:X\to Z.} Iff:X→Y{\displaystyle f:X\to Y} is continuous and

The possible topologies on a fixed setX arepartially ordered: a topologyτ1{\displaystyle \tau _{1}} is said to becoarser than another topologyτ2{\displaystyle \tau _{2}} (notation:τ1⊆τ2{\displaystyle \tau _{1}\subseteq \tau _{2}}) if every open subset with respect toτ1{\displaystyle \tau _{1}} is also open with respect toτ2.{\displaystyle \tau _{2}.} Then, theidentity mapidX:(X,τ2)→(X,τ1){\displaystyle \operatorname {id} _{X}:\left(X,\tau _{2}\right)\to \left(X,\tau _{1}\right)}is continuous if and only ifτ1⊆τ2{\displaystyle \tau _{1}\subseteq \tau _{2}} (see alsocomparison of topologies). More generally, a continuous function(X,τX)→(Y,τY){\displaystyle \left(X,\tau _{X}\right)\to \left(Y,\tau _{Y}\right)}stays continuous if the topologyτY{\displaystyle \tau _{Y}} is replaced by acoarser topology and/orτX{\displaystyle \tau _{X}} is replaced by afiner topology.

Homeomorphisms

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Symmetric to the concept of a continuous map is anopen map, for whichimages of open sets are open. If an open mapf has aninverse function, that inverse is continuous, and if a continuous mapg has an inverse, that inverse is open. Given abijective functionf between two topological spaces, the inverse functionf−1{\displaystyle f^{-1}} need not be continuous. A bijective continuous function with a continuous inverse function is called ahomeomorphism.

If a continuous bijection has as itsdomain acompact space and its codomain isHausdorff, then it is a homeomorphism.

Defining topologies via continuous functions

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Given a functionf:X→S,{\displaystyle f:X\to S,}whereX is a topological space andS is a set (without a specified topology), thefinal topology onS is defined by letting the open sets ofS be those subsetsA ofS for whichf−1(A){\displaystyle f^{-1}(A)} is open inX. IfS has an existing topology,f is continuous with respect to this topology if and only if the existing topology iscoarser than the final topology onS. Thus, the final topology is the finest topology onS that makesf continuous. Iff issurjective, this topology is canonically identified with thequotient topology under theequivalence relation defined byf.

Dually, for a functionf from a setS to a topological spaceX, theinitial topology onS is defined by designating as an open set every subsetA ofS such thatA=f−1(U){\displaystyle A=f^{-1}(U)} for some open subsetU ofX. IfS has an existing topology,f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology onS. Thus, the initial topology is the coarsest topology onS that makesf continuous. Iff is injective, this topology is canonically identified with thesubspace topology ofS, viewed as a subset ofX.

A topology on a setS is uniquely determined by the class of all continuous functionsS→X{\displaystyle S\to X} into all topological spacesX.Dually, a similar idea can be applied to mapsX→S.{\displaystyle X\to S.}

Related notions

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Iff:S→Y{\displaystyle f:S\to Y} is a continuous function from some subsetS{\displaystyle S} of a topological spaceX{\displaystyle X} then acontinuous extension off{\displaystyle f} toX{\displaystyle X} is any continuous functionF:X→Y{\displaystyle F:X\to Y} such thatF(s)=f(s){\displaystyle F(s)=f(s)} for everys∈S,{\displaystyle s\in S,} which is a condition that often written asf=F|S.{\displaystyle f=F{\big \vert }_{S}.} In words, it is any continuous functionF:X→Y{\displaystyle F:X\to Y} thatrestricts tof{\displaystyle f} onS.{\displaystyle S.} This notion is used, for example, in theTietze extension theorem and theHahn–Banach theorem. Iff:S→Y{\displaystyle f:S\to Y} is not continuous, then it could not possibly have a continuous extension. IfY{\displaystyle Y} is aHausdorff space andS{\displaystyle S} is adense subset ofX{\displaystyle X} then a continuous extension off:S→Y{\displaystyle f:S\to Y} toX,{\displaystyle X,} if one exists, will be unique. TheBlumberg theorem states that iff:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } is an arbitrary function then there exists a dense subsetD{\displaystyle D} ofR{\displaystyle \mathbb {R} } such that the restrictionf|D:D→R{\displaystyle f{\big \vert }_{D}:D\to \mathbb {R} } is continuous; in other words, every functionR→R{\displaystyle \mathbb {R} \to \mathbb {R} } can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, inorder theory, an order-preserving functionf:X→Y{\displaystyle f:X\to Y} between particular types ofpartially ordered setsX{\displaystyle X} andY{\displaystyle Y} is continuous if for eachdirected subsetA{\displaystyle A} ofX,{\displaystyle X,} we havesupf(A)=f(supA).{\displaystyle \sup f(A)=f(\sup A).} Heresup{\displaystyle \,\sup \,} is thesupremum with respect to the orderings inX{\displaystyle X} andY,{\displaystyle Y,} respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given theScott topology.[19][20]

Incategory theory, afunctorF:C→D{\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}between twocategories is calledcontinuous if it commutes with smalllimits. That is to say,lim←i∈I⁡F(Ci)≅F(lim←i∈I⁡Ci){\displaystyle \varprojlim _{i\in I}F(C_{i})\cong F\left(\varprojlim _{i\in I}C_{i}\right)}for any small (that is, indexed by a setI,{\displaystyle I,} as opposed to aclass)diagram ofobjects inC{\displaystyle {\mathcal {C}}}.

Acontinuity space is a generalization of metric spaces and posets,[21][22] which uses the concept ofquantales, and that can be used to unify the notions of metric spaces anddomains.[23]

Inmeasure theory, a functionf:E→Rk{\displaystyle f:E\to \mathbb {R} ^{k}} defined on aLebesgue measurable setE⊆Rn{\displaystyle E\subseteq \mathbb {R} ^{n}} is calledapproximately continuous at a pointx0∈E{\displaystyle x_{0}\in E} if theapproximate limit off{\displaystyle f} atx0{\displaystyle x_{0}} exists and equalsf(x0){\displaystyle f(x_{0})}. This generalizes the notion of continuity by replacing the ordinary limit with theapproximate limit. A fundamental result known as theStepanov-Denjoy theorem states that a function ismeasurable if and only if it is approximately continuousalmost everywhere.[24]

See also

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References

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Wikimedia Commons has media related toContinuity (functions).
  1. ^Bolzano, Bernard (1817)."Rein analytischer Beweis des Lehrsatzes daß zwischen je zwey Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege". Prague: Haase.
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  4. ^Jordan, M.C. (1893),Cours d'analyse de l'École polytechnique, vol. 1 (2nd ed.), Paris: Gauthier-Villars, p. 46
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  8. ^Speck, Jared (2014)."Continuity and Discontinuity"(PDF).MIT Math. p. 3. Archived fromthe original(PDF) on 2016-10-06. Retrieved2016-09-02.Example 5. The function1/x{\displaystyle 1/x} is continuous on(0,∞){\displaystyle (0,\infty )} and on(−∞,0),{\displaystyle (-\infty ,0),}, i.e., forx>0{\displaystyle x>0} and forx<0,{\displaystyle x<0,} in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namelyx=0,{\displaystyle x=0,}, and an infinite discontinuity there.
  9. ^Lang, Serge (1997),Undergraduate analysis,Undergraduate Texts in Mathematics (2nd ed.), Berlin, New York:Springer-Verlag,ISBN 978-0-387-94841-6, section II.4
  10. ^Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
  11. ^Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177
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  15. ^Searcóid, Mícheál Ó (2006),Metric spaces, Springer undergraduate mathematics series, Berlin, New York:Springer-Verlag,ISBN 978-1-84628-369-7, section 9.4
  16. ^abcDugundji 1966, pp. 211–221.
  17. ^Shurman, Jerry (2016).Calculus and Analysis in Euclidean Space (illustrated ed.). Springer. pp. 271–272.ISBN 978-3-319-49314-5.
  18. ^"general topology - Continuity and interior".Mathematics Stack Exchange.
  19. ^Goubault-Larrecq, Jean (2013).Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology.Cambridge University Press.ISBN 978-1107034136.
  20. ^Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003).Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press.ISBN 0521803381.
  21. ^Flagg, R. C. (1997). "Quantales and continuity spaces".Algebra Universalis.37 (3):257–276.CiteSeerX 10.1.1.48.851.doi:10.1007/s000120050018.S2CID 17603865.
  22. ^Kopperman, R. (1988). "All topologies come from generalized metrics".American Mathematical Monthly.95 (2):89–97.doi:10.2307/2323060.JSTOR 2323060.
  23. ^Flagg, B.; Kopperman, R. (1997)."Continuity spaces: Reconciling domains and metric spaces".Theoretical Computer Science.177 (1):111–138.doi:10.1016/S0304-3975(97)00236-3.
  24. ^Federer, H. (1969).Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag.

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