Continuous functions are of utmost importance inmathematics, functions and applications. However, not allfunctions are continuous. If a function is not continuous at alimit point (also called "accumulation point" or "cluster point") of itsdomain, one says that it has adiscontinuity there. Theset of all points of discontinuity of a function may be adiscrete set, adense set, or even the entire domain of the function.

Theoscillation of a function at a point quantifies these discontinuities as follows:

• in aremovable discontinuity, the distance that the value of the function is off by is the oscillation;
• in ajump discontinuity, the size of the jump is the oscillation (assuming that the valueat the point lies between these limits of the two sides);
• in anessential discontinuity (a.k.a. infinite discontinuity), oscillation measures the failure of alimit to exist.

A special case is if the function diverges toinfinity or minusinfinity, in which case theoscillation is not defined (in theextended real numbers, this is a removable discontinuity).

For each of the following, consider areal valued function${\displaystyle f}$ of a real variable${\displaystyle x,}$ defined in a neighborhood of the point${\displaystyle x_0}$ at which${\displaystyle f}$ is discontinuous.

Consider thepiecewise function

The point${\displaystyle x_0=1}$ is aremovablediscontinuity. For this kind of discontinuity:

Theone-sided limit from the negative direction:$${\displaystyle L^{-}=\underset{x\to x_0^{-}}{lim}f(x)}$$and the one-sided limit from the positive direction:$${\displaystyle L^{+}=\underset{x\to x_0^{+}}{lim}f(x)}$$at${\displaystyle x_0}$both exist, are finite, and are equal to${\displaystyle L=L^{-}=L^{+}.}$ In other words, since the two one-sided limits exist and are equal, the limit${\displaystyle L}$ of${\displaystyle f(x)}$ as${\displaystyle x}$ approaches${\displaystyle x_0}$ exists and is equal to this same value. If the actual value of${\displaystyle f\left(x_0\right)}$ isnot equal to${\displaystyle L,}$ then${\displaystyle x_0}$ is called aremovable discontinuity. This discontinuity can be removed to make${\displaystyle f}$ continuous at${\displaystyle x_0,}$ or more precisely, the functionis continuous at${\displaystyle x=x_0.}$

The termremovable discontinuity is sometimes broadened to include aremovable singularity, in which the limits in both directions exist and are equal, while the function isundefined at the point${\displaystyle x_0.}$^[a] This use is anabuse of terminology becausecontinuity and discontinuity of a function are concepts defined only for points in the function's domain.

Consider the function

Then, the point${\displaystyle x_0=1}$ is ajump discontinuity.

In this case, a single limit does not exist because the one-sided limits,${\displaystyle L^{-}}$ and${\displaystyle L^{+}}$ exist and are finite, but arenot equal: since,${\displaystyle L^{-}\ne L^{+},}$ the limit${\displaystyle L}$ does not exist. Then,${\displaystyle x_0}$ is called ajump discontinuity,step discontinuity, ordiscontinuity of the first kind. For this type of discontinuity, the function${\displaystyle f}$ may have any value at${\displaystyle x_0.}$

For an essential discontinuity, at least one of the two one-sided limits does not exist in${\displaystyle \mathbb{R}}$. (Notice that one or both one-sided limits can be${\displaystyle \pm \mathrm{\infty }}$).

Consider the function

Then, the point${\displaystyle x_0=1}$ is anessential discontinuity.

In this example, both${\displaystyle L^{-}}$ and${\displaystyle L^{+}}$ do not exist in${\displaystyle \mathbb{R}}$, thus satisfying the condition of essential discontinuity. So${\displaystyle x_0}$ is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from anessential singularity, which is often used when studyingfunctions of complex variables).

Supposing that${\displaystyle f}$ is a function defined on an interval${\displaystyle I\subseteq \mathbb{R},}$ we will denote by${\displaystyle D}$ the set of all discontinuities of${\displaystyle f}$ on${\displaystyle I.}$ By${\displaystyle R}$ we will mean the set of all${\displaystyle x_0\in I}$ such that${\displaystyle f}$ has aremovable discontinuity at${\displaystyle x_0.}$ Analogously by${\displaystyle J}$ we denote the set constituted by all${\displaystyle x_0\in I}$ such that${\displaystyle f}$ has ajump discontinuity at${\displaystyle x_0.}$ The set of all${\displaystyle x_0\in I}$ such that${\displaystyle f}$ has anessential discontinuity at${\displaystyle x_0}$ will be denoted by${\displaystyle E.}$ Of course then${\displaystyle D=R\cup J\cup E.}$

The two following properties of the set${\displaystyle D}$ are relevant in the literature.

• The set${\displaystyle D}$ is an${\displaystyle F_{\sigma }}$ set. The set of points at which a function is continuous is always a${\displaystyle G_{\delta }}$ set (see^[2]).
• If on the interval${\displaystyle I,}$${\displaystyle f}$ is monotone then${\displaystyle D}$ isat most countable and${\displaystyle D=J.}$ This isFroda's theorem.

Tom Apostol^[3] follows partially the classification above by considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin^[4] and Karl R. Stromberg^[5] study also removable and jump discontinuities by using different terminologies. However, furtherly, both authors state that${\displaystyle R\cup J}$ is always a countable set (see^[6][7]).

The termessential discontinuity has evidence of use in mathematical context as early as 1889.^[8] However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert.^[9] Therein, Klippert also classified essential discontinuities themselves by subdividing the set${\displaystyle E}$ into the three following sets:

$${\displaystyle E_1=\left\{x_0\in I:\underset{x\to x_0^{-}}{lim}f(x)\text{ and }\underset{x\to x_0^{+}}{lim}f(x)\text{ do not exist in }\mathbb{R}\right\},}$$$${\displaystyle E_2=\left\{x_0\in I:\underset{x\to x_0^{-}}{lim}f(x)\text{ exists in }\mathbb{R}\text{ and }\underset{x\to x_0^{+}}{lim}f(x)\text{ does not exist in }\mathbb{R}\right\},}$$$${\displaystyle E_3=\left\{x_0\in I:\underset{x\to x_0^{-}}{lim}f(x)\text{ does not exist in }\mathbb{R}\text{ and }\underset{x\to x_0^{+}}{lim}f(x)\text{ exists in }\mathbb{R}\right\}.}$$

Of course${\displaystyle E=E_1\cup E_2\cup E_3.}$ Whenever${\displaystyle x_0\in E_1,}$${\displaystyle x_0}$ is called anessential discontinuity of first kind. Any${\displaystyle x_0\in E_2\cup E_3}$ is said anessential discontinuity of second kind. Hence he enlarges the set${\displaystyle R\cup J}$ without losing its characteristic of being countable, by stating the following:

• The set${\displaystyle R\cup J\cup E_2\cup E_3}$ is countable.

When${\displaystyle I=[a,b]}$ and${\displaystyle f}$ is abounded function, it is well-known of the importance of the set${\displaystyle D}$ in the regard of the Riemannintegrability of${\displaystyle f.}$ In fact,Lebesgue's theorem (also named Lebesgue-Vitali) theorem) states that${\displaystyle f}$ is Riemann integrable on${\displaystyle I=[a,b]}$ if and only if${\displaystyle D}$ is a set with Lebesgue's measure zero.

In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function${\displaystyle f}$ be Riemann integrable on${\displaystyle [a,b].}$ Since countable sets are sets of Lebesgue's measure zero and a countable union of sets with Lebesgue's measure zero is still a set of Lebesgue's mesure zero, we are seeing now that this is not the case. In fact, the discontinuities in the set${\displaystyle R\cup J\cup E_2\cup E_3}$ are absolutely neutral in the regard of the Riemann integrability of${\displaystyle f.}$ The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows:

• A bounded function,${\displaystyle f,}$ is Riemann integrable on${\displaystyle [a,b]}$if and only if the correspondent set${\displaystyle E_1}$ of all essential discontinuities of first kind of${\displaystyle f}$ has Lebesgue's measure zero.

The case where${\displaystyle E_1=\varnothing }$ correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function${\displaystyle f:[a,b]\to \mathbb{R}}$:

• If${\displaystyle f}$ has right-hand limit at each point of${\displaystyle [a,b[}$ then${\displaystyle f}$ is Riemann integrable on${\displaystyle [a,b]}$ (see^[10])
• If${\displaystyle f}$ has left-hand limit at each point of${\displaystyle ]a,b]}$ then${\displaystyle f}$ is Riemann integrable on${\displaystyle [a,b].}$
• If${\displaystyle f}$ is aregulated function on${\displaystyle [a,b]}$ then${\displaystyle f}$ isRiemann integrable on${\displaystyle [a,b].}$

Thomae's function is discontinuous at every non-zerorational point, but continuous at everyirrational point. One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at everyrational point, but discontinuous at every irrational point.

Theindicator function of the rationals, also known as theDirichlet function, isdiscontinuous everywhere. These discontinuities are all essential of the first kind too.

Consider now the ternaryCantor set${\displaystyle \mathcal{C}\subset [0,1]}$ and its indicator (or characteristic) functionOne way to construct the Cantor set${\displaystyle \mathcal{C}}$ is given by$\mathcal{C}:=\bigcap _{n=0}^{\mathrm{\infty }}{C}_n$ where the sets${\displaystyle C_n}$ are obtained by recurrence according to$${\displaystyle C_n=\frac{C_{n-1}}{3}\cup \left(\frac{2}{3}+\frac{C_{n-1}}{3}\right)\text{ for }n\ge 1,\text{ and }{C}_0=[0,1].}$$

In view of the discontinuities of the function${\displaystyle {\mathbf{1}}_{\mathcal{C}}(x),}$ let's assume a point${\displaystyle x_0\notin \mathcal{C}.}$

Therefore there exists a set${\displaystyle C_n,}$ used in theformulation of${\displaystyle \mathcal{C}}$, which does not contain${\displaystyle x_0.}$ That is,${\displaystyle x_0}$ belongs to one of the open intervals which were removed in the construction of${\displaystyle C_n.}$ This way,${\displaystyle x_0}$ has a neighbourhood with no points of${\displaystyle \mathcal{C}.}$ (In another way, the same conclusion follows taking into account that${\displaystyle \mathcal{C}}$ is aclosed set and so its complementary with respect to${\displaystyle [0,1]}$ is open). Therefore${\displaystyle {\mathbf{1}}_{\mathcal{C}}}$ only assumes the value zero in some neighbourhood of${\displaystyle x_0.}$ Hence${\displaystyle {\mathbf{1}}_{\mathcal{C}}}$ is continuous at${\displaystyle x_0.}$

This means that the set${\displaystyle D}$ of all discontinuities of${\displaystyle {\mathbf{1}}_{\mathcal{C}}}$ on the interval${\displaystyle [0,1]}$ is a subset of${\displaystyle \mathcal{C}.}$ Since${\displaystyle \mathcal{C}}$ is anuncountable set with null Lebesgue measure, also${\displaystyle D}$ is a null Lebesgue measure set and so in the regard ofLebesgue-Vitali theorem${\displaystyle {\mathbf{1}}_{\mathcal{C}}}$ is a Riemann integrable function.

More precisely one has${\displaystyle D=\mathcal{C}.}$ In fact, since${\displaystyle \mathcal{C}}$ is a nonwhere dense set, if${\displaystyle x_0\in \mathcal{C}}$ then noneighbourhood${\displaystyle \left(x_0-\epsilon ,x_0+\epsilon \right)}$ of${\displaystyle x_0,}$ can be contained in${\displaystyle \mathcal{C}.}$ This way, any neighbourhood of${\displaystyle x_0\in \mathcal{C}}$ contains points of${\displaystyle \mathcal{C}}$ and points which are not of${\displaystyle \mathcal{C}.}$ In terms of the function${\displaystyle {\mathbf{1}}_{\mathcal{C}}}$ this means that both$\underset{x\to x_0^{-}}{lim}{\mathbf{1}}_{\mathcal{C}}(x)$ and$\underset{x\to x_0^{+}}{lim}{1}_{\mathcal{C}}(x)$ do not exist. That is,${\displaystyle D=E_1,}$ where by${\displaystyle E_1,}$ as before, we denote the set of all essential discontinuities of first kind of the function${\displaystyle {\mathbf{1}}_{\mathcal{C}}.}$ Clearly${\int }_0^1{\mathbf{1}}_{\mathcal{C}}(x)dx=0.$

Let${\displaystyle I\subseteq \mathbb{R}}$ an open interval, let${\displaystyle F:I\to \mathbb{R}}$ be differentiable on${\displaystyle I,}$ and let${\displaystyle f:I\to \mathbb{R}}$ be the derivative of${\displaystyle F.}$ That is,${\displaystyle F^{\prime }(x)=f(x)}$ for every${\displaystyle x\in I}$.According toDarboux's theorem, the derivative function${\displaystyle f:I\to \mathbb{R}}$ satisfies the intermediate value property.The function${\displaystyle f}$ can, of course, be continuous on the interval${\displaystyle I,}$ in which caseBolzano's theorem also applies. Recall that Bolzano's theorem asserts that every continuous function satisfies the intermediate value property.On the other hand, the converse is false: Darboux's theorem does not assume${\displaystyle f}$ to be continuous and the intermediate value property does not imply${\displaystyle f}$ is continuous on${\displaystyle I.}$

Darboux's theorem does, however, have an immediate consequence on the type of discontinuities that${\displaystyle f}$ can have. In fact, if${\displaystyle x_0\in I}$ is a point of discontinuity of${\displaystyle f}$, then necessarily${\displaystyle x_0}$ is an essential discontinuity of${\displaystyle f}$.^[11]This means in particular that the following two situationscannot occur:

Furthermore, two other situations have to beexcluded (see John Klippert^[12]):

Observe that whenever one of the conditions (i), (ii), (iii), or (iv) is fulfilled for some${\displaystyle x_0\in I}$ one can conclude that${\displaystyle f}$ fails to possess anantiderivative,${\displaystyle F}$, on the interval${\displaystyle I}$.

On the other hand, a new type of discontinuity with respect to any function${\displaystyle f:I\to \mathbb{R}}$ can be introduced: an essential discontinuity,${\displaystyle x_0\in I}$, of the function${\displaystyle f}$, is said to be afundamental essential discontinuity of${\displaystyle f}$ if

$${\displaystyle \underset{x\to x_0^{-}}{lim}f(x)\ne \pm \mathrm{\infty }}$$ and$${\displaystyle \underset{x\to x_0^{+}}{lim}f(x)\ne \pm \mathrm{\infty }.}$$

Therefore if${\displaystyle x_0\in I}$ is a discontinuity of a derivative function${\displaystyle f:I\to \mathbb{R}}$, then necessarily${\displaystyle x_0}$ is a fundamental essential discontinuity of${\displaystyle f}$.

Notice also that when${\displaystyle I=[a,b]}$ and${\displaystyle f:I\to \mathbb{R}}$ is a bounded function, as in the assumptions of Lebesgue's theorem, we have for all${\displaystyle x_0\in (a,b)}$:$${\displaystyle \underset{x\to x_0^{\pm }}{lim}f(x)\ne \pm \mathrm{\infty },}$$$${\displaystyle \underset{x\to a^{+}}{lim}f(x)\ne \pm \mathrm{\infty },}$$ and$${\displaystyle \underset{x\to b^{-}}{lim}f(x)\ne \pm \mathrm{\infty }.}$$Therefore any essential discontinuity of${\displaystyle f}$ is a fundamental one.

• Removable singularity – Undefined point on a holomorphic function which can be made regular
• Mathematical singularity – Point where a mathematical object behaves irregularlyPages displaying short descriptions of redirect targets
• Extension by continuity – Property of topological space
• Smoothness – Number of derivatives of a function (mathematics)
  • Geometric continuity – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets
  • Parametric continuity – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets

• Malik, S.C.; Arora, Savita (1992).Mathematical Analysis (2nd ed.). New York: Wiley.ISBN 0-470-21858-4.{{cite book}}: CS1 maint: publisher location (link)

• "Discontinuous".PlanetMath.
• "Discontinuity" byEd Pegg, Jr.,The Wolfram Demonstrations Project, 2007.
• Weisstein, Eric W."Discontinuity".MathWorld.
• Kudryavtsev, L.D. (2001) [1994]."Discontinuity point".Encyclopedia of Mathematics.EMS Press.

• Theory of continuous functions
• Mathematical analysis
• Mathematical classification systems

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