Inmathematical analysis, and applications ingeometry,applied mathematics,engineering, andnatural sciences, afunction of a real variable is afunction whosedomain is thereal numbers${\displaystyle \mathbb{R}}$, or asubset of${\displaystyle \mathbb{R}}$ that contains aninterval of positive length. Most real functions that are considered and studied aredifferentiable in some interval.The most widely considered such functions are thereal functions, which are thereal-valued functions of a real variable, that is, the functions of a real variable whosecodomain is the set of real numbers.

Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of${\displaystyle \mathbb{R}}$-vector space over the reals. That is, the codomain may be aEuclidean space, acoordinate vector, the set ofmatrices of real numbers of a given size, or an${\displaystyle \mathbb{R}}$-algebra, such as thecomplex numbers or thequaternions. The structure${\displaystyle \mathbb{R}}$-vector space of the codomain induces a structure of${\displaystyle \mathbb{R}}$-vector space on the functions. If the codomain has a structure of${\displaystyle \mathbb{R}}$-algebra, the same is true for the functions.

Theimage of a function of a real variable is acurve in the codomain. In this context, a function that defines curve is called aparametric equation of the curve.

When the codomain of a function of a real variable is afinite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.

A real function is afunction from a subset of${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R},}$ where${\displaystyle \mathbb{R}}$ denotes as usual the set ofreal numbers. That is, thedomain of a real function is a subset${\displaystyle \mathbb{R}}$, and itscodomain is${\displaystyle \mathbb{R}.}$ It is generally assumed that the domain contains aninterval of positive length.

For many commonly used real functions, the domain is the whole set of real numbers, and the function iscontinuous anddifferentiable at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:

• Allpolynomial functions, includingconstant functions andlinear functions
• Sine andcosine functions
• Exponential function

Some functions are defined everywhere, but not continuous at some points. For example

• TheHeaviside step function is defined everywhere, but not continuous at zero.

Some functions are defined and continuous everywhere, but not everywhere differentiable. For example

• Theabsolute value is defined and continuous everywhere, and is differentiable everywhere, except for zero.
• Thecubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero.

Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:

• Arational function is a quotient of two polynomial functions, and is not defined at thezeros of the denominator.
• Thetangent function is not defined for${\displaystyle \frac{\pi }{2}+k\pi ,}$ wherek is any integer.
• Thelogarithm function is defined only for positive values of the variable.

Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:

• Thesquare root is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).

Areal-valued function of a real variable is afunction that takes as input areal number, commonly represented by thevariablex, for producing another real number, thevalue of the function, commonly denotedf(x). For simplicity, in this article a real-valued function of a real variable will be simply called afunction. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.

Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subsetX of${\displaystyle \mathbb{R}}$, thedomain of the function, which is always supposed to contain aninterval of positive length. In other words, a real-valued function of a real variable is a function

${\displaystyle f:X\to \mathbb{R}}$

such that its domainX is a subset of${\displaystyle \mathbb{R}}$ that contains an interval of positive length.

A simple example of a function in one variable could be:

${\displaystyle f:X\to \mathbb{R}}$

${\displaystyle X=\{x\in \mathbb{R}:x\ge 0\}}$

${\displaystyle f(x)=\sqrt{x}}$

which is thesquare root ofx.

Theimage of a function${\displaystyle f(x)}$ is the set of all values off when the variablex runs in the whole domain off. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either aninterval or a single value. In the latter case, the function is aconstant function.

Thepreimage of a given real numbery is the set of the solutions of theequationy =f(x).

Thedomain of a function of several real variables is a subset of${\displaystyle \mathbb{R}}$ that is sometimes explicitly defined. In fact, if one restricts the domainX of a functionf to a subsetY ⊂X, one gets formally a different function, therestriction off toY, which is denotedf_|Y. In practice, it is often not harmful to identifyf andf_|Y, and to omit the subscript_|Y.

Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example bycontinuity or byanalytic continuation. This means that it is not worthy to explicitly define the domain of a function of a real variable.

The arithmetic operations may be applied to the functions in the following way:

• For every real numberr, theconstant function${\displaystyle (x)\mapsto r}$, is everywhere defined.
• For every real numberr and every functionf, the function${\displaystyle rf:(x)\mapsto rf(x)}$ has the same domain asf (or is everywhere defined ifr = 0).
• Iff andg are two functions of respective domainsX andY such thatX∩Y contains an open subset of${\displaystyle \mathbb{R}}$, then${\displaystyle f+g:(x)\mapsto f(x)+g(x)}$ and${\displaystyle fg:(x)\mapsto f(x)g(x)}$ are functions that have a domain containingX∩Y.

It follows that the functions ofn variables that are everywhere defined and the functions ofn variables that are defined in someneighbourhood of a given point both formcommutative algebras over the reals (${\displaystyle \mathbb{R}}$-algebras).

One may similarly define${\displaystyle 1/f:(x)\mapsto 1/f(x),}$ which is a function only if the set of the points(x) in the domain off such thatf(x) ≠ 0 contains an open subset of${\displaystyle \mathbb{R}}$. This constraint implies that the above two algebras are notfields.

Until the second part of 19th century, onlycontinuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of atopological space and acontinuous map between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.

For defining the continuity, it is useful to consider thedistance function of${\displaystyle \mathbb{R}}$, which is an everywhere defined function of 2 real variables:${\displaystyle d(x,y)=|x-y|}$

A functionf iscontinuous at a point${\displaystyle a}$ which isinterior to its domain, if, for every positive real numberε, there is a positive real numberδ such that for all${\displaystyle x}$ such that In other words,δ may be chosen small enough for having the image byf of the interval of radiusδ centered at${\displaystyle a}$ contained in the interval of length2ε centered at${\displaystyle f(a).}$ A function is continuous if it is continuous at every point of its domain.

Thelimit of a real-valued function of a real variable is as follows.^[1] Leta be a point intopological closure of the domainX of the functionf. The function,f has a limitL whenx tends towarda, denoted

${\displaystyle L=\underset{x\to a}{lim}f(x),}$

if the following condition is satisfied:For every positive real numberε > 0, there is a positive real numberδ > 0 such that

for allx in the domain such that

If the limit exists, it is unique. Ifa is in the interior of the domain, the limit exists if and only if the function is continuous ata. In this case, we have

${\displaystyle f(a)=\underset{x\to a}{lim}f(x).}$

Whena is in theboundary of the domain off, and iff has a limit ata, the latter formula allows to "extend by continuity" the domain off toa.

One can collect a number of functions each of a real variable, say

${\displaystyle y_1=f_1(x),y_2=f_2(x),\dots ,y_n=f_n(x)}$

into a vector parametrized byx:

${\displaystyle \mathbf{y}=(y_1,y_2,\dots ,y_n)=[f_1(x),f_2(x),\dots ,f_n(x)]}$

The derivative of the vectory is the vector derivatives off_i(x) fori = 1, 2, ...,n:

${\displaystyle \frac{d\mathbf{y}}{dx}=\left(\frac{d{y}_1}{dx},\frac{d{y}_2}{dx},\dots ,\frac{d{y}_n}{dx}\right)}$

One can also performline integrals along aspace curve parametrized byx, withposition vectorr =r(x), by integrating with respect to the variablex:

${\displaystyle {\int }_a^b\mathbf{y}(x)\cdot d\mathbf{r}={\int }_a^b\mathbf{y}(x)\cdot \frac{d\mathbf{r}(x)}{dx}dx}$

where · is thedot product, andx =a andx =b are the start and endpoints of the curve.

With the definitions of integration and derivatives, key theorems can be formulated, including thefundamental theorem of calculus,integration by parts, andTaylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theoremdifferentiation under the integral sign.

Areal-valuedimplicit function of a real variable is not written in the form "y =f(x)". Instead, the mapping is from the space${\displaystyle \mathbb{R}}$² to thezero element in${\displaystyle \mathbb{R}}$ (just the ordinary zero 0):

${\displaystyle \varphi :{\mathbb{R}}^2\to \{0\}}$

and

${\displaystyle \varphi (x,y)=0}$

is an equation in the variables. Implicit functions are a more general way to represent functions, since if:

${\displaystyle y=f(x)}$

then we can always define:

${\displaystyle \varphi (x,y)=y-f(x)=0}$

but the converse is not always possible, i.e. not all implicit functions have the form of this equation.

Given the functionsr₁ =r₁(t),r₂ =r₂(t), ...,r_n =r_n(t) all of a common variablet, so that:

or taken together:

${\displaystyle \mathbf{r}:\mathbb{R}\to {\mathbb{R}}^n,\mathbf{r}=\mathbf{r}(t)}$

then the parametrizedn-tuple,

${\displaystyle \mathbf{r}(t)=[r_1(t),r_2(t),\dots ,r_n(t)]}$

describes a one-dimensionalspace curve.

At a pointr(t =c) =a = (a₁,a₂, ...,a_n) for some constantt =c, the equations of the one-dimensional tangent line to the curve at that point are given in terms of theordinary derivatives ofr₁(t),r₂(t), ...,r_n(t), andr with respect tot:

${\displaystyle \frac{r_1(t)-a_1}{d{r}_1(t)/dt}=\frac{r_2(t)-a_2}{d{r}_2(t)/dt}=\cdots =\frac{r_n(t)-a_n}{d{r}_n(t)/dt}}$

The equation of then-dimensionalhyperplane normal to the tangent line atr =a is:

${\displaystyle (p_1-a_1)\frac{d{r}_1(t)}{dt}+(p_2-a_2)\frac{d{r}_2(t)}{dt}+\cdots +(p_n-a_n)\frac{d{r}_n(t)}{dt}=0}$

or in terms of thedot product:

${\displaystyle (\mathbf{p}-\mathbf{a})\cdot \frac{d\mathbf{r}(t)}{dt}=0}$

wherep = (p₁,p₂, ...,p_n) are pointsin the plane, not on the space curve.

The physical and geometric interpretation ofdr(t)/dt is the "velocity" of a point-likeparticle moving along the pathr(t), treatingr as the spatialposition vector coordinates parametrized by timet, and is a vector tangent to the space curve for allt in the instantaneous direction of motion. Att =c, the space curve has a tangent vectordr(t)/dt|_{t =c}, and the hyperplane normal to the space curve att =c is also normal to the tangent att =c. Any vector in this plane (p −a) must be normal todr(t)/dt|_{t =c}.

Similarly,d²r(t)/dt² is the "acceleration" of the particle, and is a vector normal to the curve directed along theradius of curvature.

Amatrix can also be a function of a single variable. For example, therotation matrix in 2d:

is a matrix valued function of rotation angle of about the origin. Similarly, inspecial relativity, theLorentz transformation matrix for a pure boost (without rotations):

is a function of the boost parameterβ =v/c, in whichv is therelative velocity between the frames of reference (a continuous variable), andc is thespeed of light, a constant.

Generalizing the previous section, the output of a function of a real variable can also lie in aBanach space or aHilbert space. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of aket or anoperator. This occurs, for instance, in the general time-dependentSchrödinger equation:

${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }=\hat{H}\mathrm{\Psi }}$

where one takes the derivative of a wave function, which can be an element of several different Hilbert spaces.

Acomplex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowingcomplex values.

Iff(x) is such a complex valued function, it may be decomposed as

f(x) =g(x) +ih(x),

whereg andh are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.

Thecardinality of the set of real-valued functions of a real variable,${\displaystyle {\mathbb{R}}^{\mathbb{R}}=\{f:\mathbb{R}\to \mathbb{R}\}}$, is${\displaystyle {\beth }_2=2^{\mathfrak{c}}}$, which is strictly larger than the cardinality of thecontinuum (i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic:

$${\displaystyle \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}({\mathbb{R}}^{\mathbb{R}})=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{R}{)}^{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{R})}={\mathfrak{c}}^{\mathfrak{c}}=(2^{{\mathrm{\aleph }}_0}{)}^{\mathfrak{c}}=2^{{\mathrm{\aleph }}_0\cdot \mathfrak{c}}=2^{\mathfrak{c}}.}$$

Furthermore, if${\displaystyle X}$ is a set such that${\displaystyle 2\le \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(X)\le \mathfrak{c}}$, then the cardinality of the set${\displaystyle X^{\mathbb{R}}=\{f:\mathbb{R}\to X\}}$ is also${\displaystyle 2^{\mathfrak{c}}}$, since

$${\displaystyle 2^{\mathfrak{c}}=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(2^{\mathbb{R}})\le \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(X^{\mathbb{R}})\le \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}({\mathbb{R}}^{\mathbb{R}})=2^{\mathfrak{c}}.}$$

However, the set ofcontinuous functions${\displaystyle C^0(\mathbb{R})=\{f:\mathbb{R}\to \mathbb{R}:f\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\}}$ has a strictly smaller cardinality, the cardinality of the continuum,${\displaystyle \mathfrak{c}}$. This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.^[2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:

$${\displaystyle \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(C^0(\mathbb{R}))\le \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}({\mathbb{R}}^{\mathbb{Q}})=(2^{{\mathrm{\aleph }}_0}{)}^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0\cdot {\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0}=\mathfrak{c}.}$$

On the other hand, since there is a clearbijection between${\displaystyle \mathbb{R}}$ and the set of constant functions${\displaystyle \{f:\mathbb{R}\to \mathbb{R}:f(x)\equiv x_0\}}$, which forms a subset of${\displaystyle C^0(\mathbb{R})}$,${\displaystyle \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(C^0(\mathbb{R}))\ge \mathfrak{c}}$ must also hold. Hence,${\displaystyle \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(C^0(\mathbb{R}))=\mathfrak{c}}$.

• Real analysis
• Function of several real variables
• Complex analysis
• Function of several complex variables

• F. Ayres, E. Mendelson (2009).Calculus. Schaum's outline series (5th ed.). McGraw Hill.ISBN 978-0-07-150861-2.
• R. Wrede, M. R. Spiegel (2010).Advanced calculus. Schaum's outline series (3rd ed.). McGraw Hill.ISBN 978-0-07-162366-7.
• N. Bourbaki (2004).Functions of a Real Variable: Elementary Theory. Springer.ISBN 354-065-340-6.

• Multivariable Calculus
• L. A. Talman (2007)Differentiability for Multivariable Functions

• Mathematical analysis
• Real numbers
• Multivariable calculus

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