Editing Function of a real variable

{{Short description|Mathematical function}}
{{functions}}

In [[mathematical analysis]], and applications in [[geometry]], [[applied mathematics]], [[engineering]], and [[natural science]]s, a '''function of a real variable''' is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is the [[real number]]s <math>\mathbb{R}</math>, or a [[subset]] of <math>\mathbb{R}</math> that contains an [[interval (mathematics)|interval]] of positive length. Most real functions that are considered and studied are [[differentiable function|differentiable]] in some interval.
The most widely considered such functions are the '''real functions''', which are the [[real-valued function]]s of a real variable, that is, the functions of a real variable whose [[codomain]] 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 <math>\mathbb{R}</math>-[[vector space]] over the reals. That is, the codomain may be a [[Euclidean space]], a [[coordinate vector]], the set of [[matrix (mathematics)|matrices]] of real numbers of a given size, or an <math>\mathbb{R}</math>-[[algebra over a field|algebra]], such as the [[complex number]]s or the [[quaternion]]s. The structure <math>\mathbb{R}</math>-vector space of the codomain induces a structure of <math>\mathbb{R}</math>-vector space on the functions. If the codomain has a structure of <math>\mathbb{R}</math>-algebra, the same is true for the functions.

The [[image (mathematics)|image]] of a function of a real variable is a [[curve (mathematics)|curve]] in the codomain. In this context, a function that defines curve is called a [[parametric equation]] of the curve.

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

==Real function==
[[File:Real function of one variable.svg|thumb|The graph of a real function]]
A real function is a [[function (mathematics)|function]] from a subset of <math>\mathbb R</math> to <math>\mathbb R,</math> where <math>\mathbb R</math> denotes as usual the set of [[real number]]s. That is, the [[domain of a function|domain]] of a real function is a subset <math>\mathbb R</math>, and its [[codomain]] is <math>\mathbb R.</math> It is generally assumed that the domain contains an [[interval (mathematics)|interval]] of positive length.

===Basic examples===

For many commonly used real functions, the domain is the whole set of real numbers, and the function is [[continuous function|continuous]] and [[differentiable function|differentiable]] at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:
* All [[polynomial function]]s, including [[constant function]]s and [[linear function (calculus)|linear functions]]
* [[Sine]] and [[cosine]] functions
* [[Exponential function]]

Some functions are defined everywhere, but not continuous at some points. For example
* The [[Heaviside step function]] is defined everywhere, but not continuous at zero.

Some functions are defined and continuous everywhere, but not everywhere differentiable. For example
* The [[absolute value]] is defined and continuous everywhere, and is differentiable everywhere, except for zero.
* The [[cubic 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:
* A [[rational function]] is a quotient of two polynomial functions, and is not defined at the [[zero of a function|zeros]] of the denominator.
* The [[tangent function]] is not defined for <math>\frac\pi 2 + k\pi,</math> where {{mvar|k}} is any integer.
* The [[logarithm 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:
*The [[square 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).

==General definition ==

A '''real-valued function of a real variable''' is a [[function (mathematics)|function]] that takes as input a [[real number]], commonly represented by the [[variable (mathematics)|variable]] ''x'', for producing another real number, the ''value'' of the function, commonly denoted ''f''(''x''). For simplicity, in this article a real-valued function of a real variable will be simply called a '''function'''. 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 subset ''X'' of <math>\mathbb{R}</math>, the [[domain of a function|domain]] of the function, which is always supposed to contain an [[interval (mathematics)|interval]] of positive length. In other words, a real-valued function of a real variable is a function

:<math>f: X \to \R </math>

such that its domain ''X'' is a subset of <math>\mathbb{R}</math> that contains an interval of positive length.

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

:<math> f : X \to \R </math>
:<math> X = \{ x \in \R \,:\, x \geq 0\} </math>
:<math> f(x) = \sqrt{x}</math>

which is the [[square root]] of ''x''.

===Image===
{{Main|Image (mathematics)}}
The [[image (mathematics)|image]] of a function <math>f(x)</math> is the set of all values of {{mvar|''f''}} when the variable ''x'' runs in the whole domain of {{mvar|''f''}}. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an [[interval (mathematics)|interval]] or a single value. In the latter case, the function is a [[constant function]].

The [[preimage]] of a given real number ''y'' is the set of the solutions of the [[equation]] {{nowrap|1=''y'' = ''f''(''x'')}}.

=== Domain ===

The [[Domain of a function|domain]] of a function of several real variables is a subset of <math>\mathbb{R}</math> that is sometimes explicitly defined. In fact, if one restricts the domain ''X'' of a function ''f'' to a subset ''Y'' ⊂ ''X'', one gets formally a different function, the ''restriction'' of ''f'' to ''Y'', which is denoted ''f''<sub>|''Y''</sub>. In practice, it is often not harmful to identify ''f'' and ''f''<sub>|''Y''</sub>, and to omit the subscript <sub>|''Y''</sub>.

Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by [[continuous function|continuity]] or by [[analytic continuation]]. This means that it is not worthy to explicitly define the domain of a function of a real variable.

=== Algebraic structure ===

The arithmetic operations may be applied to the functions in the following way:
* For every real number ''r'', the [[constant function]] <math>(x)\mapsto r</math>, is everywhere defined.
* For every real number ''r'' and every function ''f'', the function <math>rf:(x)\mapsto rf(x)</math> has the same domain as ''f'' (or is everywhere defined if ''r'' = 0).
* If ''f'' and ''g'' are two functions of respective domains ''X'' and ''Y'' such that {{nowrap|''X''∩''Y''}} contains an open subset of <math>\mathbb{R}</math>, then <math>f+g:(x)\mapsto f(x)+g(x)</math> and <math>f\,g:(x)\mapsto f(x)\,g(x)</math> are functions that have a domain containing {{nowrap|''X''∩''Y''}}.

It follows that the functions of ''n'' variables that are everywhere defined and the functions of ''n'' variables that are defined in some [[neighbourhood (mathematics)|neighbourhood]] of a given point both form [[commutative algebra (structure)|commutative algebras]] over the reals (<math>\mathbb{R}</math>-algebras).

One may similarly define <math>1/f:(x)\mapsto 1/f(x),</math> which is a function only if the set of the points {{nowrap|(''x'')}} in the domain of ''f'' such that {{nowrap|''f''(''x'') ≠ 0}} contains an open subset of <math>\mathbb{R}</math>. This constraint implies that the above two algebras are not [[field (mathematics)|fields]].

===Continuity and limit===
[[File:Limit of a real function of a real variable.svg|thumb|Limit of a real function of a real variable.]]
Until the second part of 19th century, only [[continuous function]]s 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 a [[topological space]] and a [[continuous 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 the [[distance function]] of <math>\mathbb{R}</math>, which is an everywhere defined function of 2 real variables:
<math>d(x,y)=|x-y|</math>

A function ''f'' is '''continuous''' at a point <math>a</math> which is [[interior (topology)|interior]] to its domain, if, for every positive real number {{math|''ε''}}, there is a positive real number {{math|''φ''}} such that <math>|f(x)-f(a)| < \varepsilon </math> for all <math>x</math> such that <math>d(x,a)<\varphi.</math> In other words, {{math|''φ''}} may be chosen small enough for having the image by ''f'' of the interval of radius {{math|''φ''}} centered at <math>a</math> contained in the interval of length {{math|2''ε''}} centered at <math>f(a).</math> A function is continuous if it is continuous at every point of its domain.

The [[limit (mathematics)|limit]] of a real-valued function of a real variable is as follows.<ref>{{cite book|title=Differential and Integral Calculus|volume=2|author=R. Courant|date=23 February 1988|pages=46–47|publisher=Wiley Classics Library|isbn=0-471-60840-8}}</ref> Let ''a'' be a point in [[closure (topology)|topological closure]] of the domain ''X'' of the function ''f''. The function, ''f'' has a limit ''L'' when ''x'' tends toward ''a'', denoted 
:<math>L = \lim_{x \to a} f(x), </math>
if the following condition is satisfied:
For every positive real number ''ε'' > 0, there is a positive real number ''δ'' > 0 such that 
:<math>|f(x) - L| < \varepsilon  </math>
for all ''x'' in the domain such that
:<math>d(x, a)< \delta.</math>

If the limit exists, it is unique. If ''a'' is in the interior of the domain, the limit exists if and only if the function is continuous at ''a''. In this case, we have 
:<math>f(a) = \lim_{x \to a} f(x). </math>

When ''a'' is in the [[boundary (topology)|boundary]] of the domain of ''f'', and if ''f'' has a limit at ''a'', the latter formula allows to "extend by continuity" the domain of ''f'' to ''a''.

==Calculus==

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

:<math>y_1 = f_1(x)\,,\quad y_2 = f_2(x)\,,\ldots, y_n = f_n(x) </math>

into a vector parametrized by ''x'':

:<math>\mathbf{y} = (y_1,  y_2, \ldots, y_n) = [f_1(x), f_2(x) ,\ldots, f_n(x)]  </math>

The derivative of the vector '''y''' is the vector derivatives of ''f<sub>i</sub>''(''x'') for ''i'' = 1, 2, ..., ''n'':

:<math>\frac{d\mathbf{y}}{dx} = \left(\frac{dy_1}{dx}, \frac{dy_2}{dx}, \ldots, \frac{dy_n}{dx}\right) </math>

One can also perform [[line integral]]s along a [[space curve]] parametrized by ''x'', with [[position vector]] '''r''' = '''r'''(''x''), by integrating with respect to the variable ''x'':

:<math>\int_a^b \mathbf{y}(x) \cdot d\mathbf{r} = \int_a^b \mathbf{y}(x) \cdot \frac{d\mathbf{r}(x)}{dx} dx </math>

where · is the [[dot product]], and ''x'' = ''a'' and ''x'' = ''b'' are the start and endpoints of the curve.

===Theorems===

With the definitions of integration and derivatives, key theorems can be formulated, including the [[fundamental theorem of calculus]], [[integration by parts]], and [[Taylor's theorem]]. Evaluating a mixture of integrals and derivatives can be done by using theorem [[differentiation under the integral sign]].

==Implicit functions==

A '''real-valued [[implicit function]] of a real variable''' is not written in the form "''y'' = ''f''(''x'')". Instead, the mapping is from the space <math>\mathbb{R}</math><sup>2</sup> to the [[zero element]] in <math>\mathbb{R}</math> (just the ordinary zero 0):

:<math>\phi: \R^2 \to \{0\} </math>

and

:<math>\phi(x,y) = 0 </math>

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

:<math>y=f(x) </math>

then we can always define:

:<math> \phi(x, y) = y - f(x) = 0 </math>

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

==One-dimensional space curves in <math>\mathbb{R}</math><sup>''n''</sup>==

[[File:Space curve.svg|200px|thumb|Space curve in 3d. The [[position vector]] '''r''' is parametrized by a scalar ''t''. At '''r''' = '''a''' the red line is the tangent to the curve, and the blue plane is normal to the curve.]]

===Formulation===

Given the functions {{nowrap|''r''<sub>1</sub> {{=}} ''r''<sub>1</sub>(''t'')}}, {{nowrap|''r''<sub>2</sub> {{=}} ''r''<sub>2</sub>(''t'')}}, ..., {{nowrap|''r''<sub>''n''</sub> {{=}} ''r''<sub>''n''</sub>(''t'')}} all of a common variable ''t'', so that:

:<math>\begin{align}
r_1 : \mathbb{R} \rightarrow \mathbb{R} & \quad r_2 : \mathbb{R} \rightarrow \mathbb{R} & \cdots &  \quad r_n : \mathbb{R} \rightarrow \mathbb{R} \\
r_1 = r_1(t) &  \quad r_2 = r_2(t) & \cdots &  \quad r_n = r_n(t) \\
\end{align}</math>

or taken together:

:<math>\mathbf{r} : \mathbb{R} \rightarrow \mathbb{R}^n \,,\quad \mathbf{r} = \mathbf{r}(t) </math>

then the parametrized ''n''-tuple,

:<math>\mathbf{r}(t) = [r_1(t), r_2(t), \ldots , r_n(t)] </math>

describes a one-dimensional [[space curve]].

===Tangent line to curve===

At a point {{nowrap|'''r'''(''t'' {{=}} ''c'') {{=}} '''a''' {{=}} (''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>)}} for some constant ''t'' = ''c'', the equations of the one-dimensional tangent line to the curve at that point are given in terms of the [[ordinary derivative]]s of ''r''<sub>1</sub>(''t''), ''r''<sub>2</sub>(''t''), ..., ''r''<sub>''n''</sub>(''t''), and ''r'' with respect to ''t'':

:<math>\frac{r_1(t) - a_1}{dr_1(t)/dt} = \frac{r_2(t) - a_2}{dr_2(t)/dt} = \cdots = \frac{r_n(t) - a_n}{dr_n(t)/dt} </math>

===Normal plane to curve===

The equation of the ''n''-dimensional [[hyperplane]] normal to the tangent line at '''r''' = '''a''' is:

:<math>(p_1 - a_1)\frac{dr_1(t)}{dt} + (p_2 - a_2)\frac{dr_2(t)}{dt} + \cdots + (p_n - a_n)\frac{dr_n(t)}{dt} = 0</math>

or in terms of the [[dot product]]:

:<math>(\mathbf{p} - \mathbf{a})\cdot \frac{d\mathbf{r}(t)}{dt} = 0</math>

where {{nowrap|'''p''' {{=}} (''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>)}} are points ''in the plane'', not on the space curve.

===Relation to kinematics===

[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass ''m'', position '''r''', velocity '''v''', acceleration '''a'''.]]

The physical and geometric interpretation of ''d'''''r'''(''t'')/''dt'' is the "[[velocity]]" of a point-like [[particle]] moving along the path '''r'''(''t''), treating '''r''' as the spatial [[position vector]] coordinates parametrized by time ''t'', and is a vector tangent to the space curve for all ''t'' in the instantaneous direction of motion. At ''t'' = ''c'', the space curve has a tangent vector {{nowrap|''d'''''r'''(''t'')/''dt''{{!}}<sub>''t'' {{=}} ''c''</sub>}}, and the hyperplane normal to the space curve at ''t'' = ''c'' is also normal to the tangent at ''t'' = ''c''. Any vector in this plane ('''p''' − '''a''') must be normal to {{nowrap|''d'''''r'''(''t'')/''dt''{{!}}<sub>''t'' {{=}} ''c''</sub>}}.

Similarly, ''d''<sup>2</sup>'''r'''(''t'')/''dt''<sup>2</sup> is the "[[acceleration]]" of the particle, and is a vector normal to the curve directed along the [[Radius of curvature (mathematics)|radius of curvature]].

==Matrix valued functions==

A [[matrix (mathematics)|matrix]] can also be a function of a single variable. For example, the [[rotation matrix]] in 2d:

:<math>
R(\theta) = \begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}</math>

is a matrix valued function of rotation angle of about the origin. Similarly, in [[special relativity]], the [[Lorentz transformation]] matrix for a pure boost (without rotations):

:<math>
\Lambda(\beta) = \begin{bmatrix}
\frac{1}{\sqrt{1-\beta ^2}} & -\frac{\beta }{\sqrt{1-\beta ^2}} & 0 & 0 \\
-\frac{\beta }{\sqrt{1-\beta ^2}} & \frac{1}{\sqrt{1-\beta ^2}} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}</math>

is a function of the boost parameter ''β'' = ''v''/''c'', in which ''v'' is the [[relative velocity]] between the frames of reference (a continuous variable), and ''c'' is the [[speed of light]], a constant.

==Banach and Hilbert spaces and quantum mechanics==
Generalizing the previous section, the output of a function of a real variable can also lie in a [[Banach space]] or a [[Hilbert 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 a [[bra–ket notation|ket]] or an [[operator (physics)|operator]]. This occurs, for instance, in the general time-dependent [[Schrödinger equation]]:

:<math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi</math>

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

== Complex-valued function of a real variable==

A '''complex-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 allowing [[complex number|complex]] values.

If {{math|''f''(''x'')}} is such a complex valued function, it may be decomposed as 
:{{math|''f''(''x'')}} = {{math|''g''(''x'')}} + {{math|''ih''(''x'')}},
where {{math|''g''}} and {{math|''h''}} 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.

== Cardinality of sets of functions of a real variable ==
The [[cardinality]] of the set of real-valued functions of a real variable, <math>\mathbb{R}^\mathbb{R}=\{f:\mathbb{R}\to \mathbb{R}\}</math>, is <math>\beth_2=2^\mathfrak{c}</math>, which is strictly larger than the cardinality of the [[Continuum (set theory)|continuum]] (i.e., set of all real numbers).  This fact is easily verified by cardinal arithmetic:

<math display="block">\mathrm{card}(\R^\R)=\mathrm{card}(\R)^{\mathrm{card}(\R)}=
\mathfrak{c}^\mathfrak{c}=(2^{\aleph_0})^\mathfrak{c}=2^{\aleph_0\cdot\mathfrak{c}}=2^\mathfrak{c}.
</math>

Furthermore, if <math>X</math> is a set such that <math>2\leq\mathrm{card}(X)\leq\mathfrak{c}</math>, then the cardinality of the set <math>X^\mathbb{R}=\{f:\mathbb{R}\to X\}</math> is also <math>2^\mathfrak{c}</math>, since

<math display="block">2^\mathfrak{c}=\mathrm{card}(2^\R)\leq\mathrm{card}(X^\R)\leq\mathrm{card}(\R^
\R)=2^\mathfrak{c}.</math>

However, the set of [[continuous function]]s <math>C^0(\mathbb{R})=\{f:\mathbb{R}\to\mathbb{R}:f\ \mathrm{continuous}\}</math> has a strictly smaller cardinality, the cardinality of the continuum, <math>\mathfrak{c}</math>.  This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.<ref>{{Cite book|title=Principles of Mathematical Analysis|last=Rudin|first=W.|publisher=McGraw-Hill|year=1976|isbn=0-07-054235X|location=New York|pages=98–99}}</ref>  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:

<math display="block">\mathrm{card}(C^0(\R))\leq\mathrm{card}(\R^\Q)=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=
2^{\aleph_0}=\mathfrak{c}.</math>

On the other hand, since there is a clear [[bijection]] between <math>\R</math> and the set of constant functions <math>\{f:\R\to\R: f(x)\equiv x_0\}</math>, which forms a subset of <math>C^0(\R)</math>, <math>\mathrm{card}(C^0(\R)) \geq \mathfrak{c}</math> must also hold.  Hence, <math>\mathrm{card}(C^0(\R)) = \mathfrak{c}</math>.

==See also==

*[[Real analysis]]
*[[Function of several real variables]]
*[[Complex analysis]]
*[[Function of several complex variables]]

==References==

{{Reflist}}

* {{cite book |author=F. Ayres, E. Mendelson| title=Calculus |edition=5th|publisher= McGraw Hill| year=2009| series=Schaum's outline series|isbn=978-0-07-150861-2}}
* {{cite book |author=R. Wrede, M. R. Spiegel| title=Advanced calculus|edition=3rd|publisher= McGraw Hill| year=2010| series=Schaum's outline series|isbn=978-0-07-162366-7}}
*{{cite book|title=Functions of a Real Variable: Elementary Theory|author=N. Bourbaki|publisher=Springer|year=2004 |isbn=354-065-340-6|url=https://books.google.com/books?id=dtYLvM02cRYC&q=real+valued+functions+of+several+real+variables&pg=PA1}}

==External links==
*[https://web.archive.org/web/20131021163843/http://math.bard.edu/belk/math461/MultivariableCalculus.pdf ''Multivariable Calculus'']
*[https://web.archive.org/web/20100620155743/http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf L. A. Talman (2007) ''Differentiability for Multivariable Functions'']

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[[Category:Mathematical analysis]]
[[Category:Real numbers]]
[[Category:Multivariable calculus]]