131.0.197.244 at 01:18, 9 June 2025

2025-06-09T01:18:14Z

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{{short description|Mathematical function whose derivative exists}}
[[File:Polynomialdeg3.svg|thumb|right|A differentiable function]]
In [[mathematics]], a '''differentiable function''' of one [[Real number|real]] variable is a [[Function (mathematics)|function]] whose [[derivative]] exists at each point in its [[Domain of a function|domain]]. In other words, the [[Graph of a function|graph]] of a differentiable function has a non-[[Vertical tangent|vertical]] [[tangent line]] at each interior point in its domain. A differentiable function is [[Smoothness|smooth]] (the function is locally well approximated as a [[linear function]] at each interior point) and does not contain any break, angle, or [[Cusp (singularity)|cusp]].

If {{math|''x''0}} is an interior point in the domain of a function {{mvar|f}}, then {{mvar|f}} is said to be ''differentiable at'' {{math|''x''0}} if the derivative <math>f'(x_0)</math> exists. In other words, the graph of {{mvar|f}} has a non-vertical tangent line at the point {{math|(''x''0, ''f''(''x''0))}}. {{mvar|f}} is said to be differentiable on {{mvar|U}} if it is differentiable at every point of {{mvar|U}}. {{mvar|f}} is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function <math display="inline">f</math>. Generally speaking, {{mvar|f}} is said to be of class {{em|<math>C^k</math>}} if its first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> exist and are continuous over the domain of the function <math display="inline">f</math>.

For a multivariable function, as shown [[#Differentiability in higher dimensions|here]], the differentiability of it is something more complex than the existence of the partial derivatives of it.

==Differentiability of real functions of one variable==

A function <math>f:U\to\mathbb{R}</math>, defined on an open set <math display="inline">U\subset\mathbb{R}</math>, is said to be ''differentiable'' at <math>a\in U</math> if the derivative
:<math>f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}</math>
exists. This implies that the function is [[continuous function|continuous]] at {{mvar|a}}.

This function {{mvar|f}} is said to be ''differentiable'' on {{mvar|U}} if it is differentiable at every point of {{mvar|U}}. In this case, the derivative of {{mvar|f}} is thus a function from {{mvar|U}} into <math>\mathbb R.</math>

A continuous function is not necessarily differentiable, but a differentiable function is necessarily [[continuous function|continuous]] (at every point where it is differentiable) as is shown below (in the section [[Differentiable function#Differentiability and continuity|Differentiability and continuity]]). A function is said to be ''continuously differentiable'' if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section [[Differentiable function#Differentiability classes|Differentiability classes]]).
===Semi-differentiability===
{{Main|Semi-differentiability}}
The above definition can be extended to define the derivative at [[Boundary (topology)|boundary points]]. The derivative of a function <math display="inline">f:A\to \mathbb{R}</math> defined on a closed subset <math display="inline">A\subsetneq \mathbb{R}</math> of the real numbers, evaluated at a boundary point <math display="inline">c</math>, can be defined as the following one-sided limit, where the argument <math display="inline">x</math> approaches <math display="inline">c</math> such that it is always within <math display="inline">A</math>:
:<math>f'(c)=\lim_{{\scriptstyle x\to c\atop\scriptstyle x\in A}}\frac{f(x)-f(c)}{x-c}.</math>

For <math display="inline">x</math> to remain within <math display="inline">A</math>, which is a subset of the reals, it follows that this limit will be defined as either
:<math>f'(c)=\lim_{x\to c^+}\frac{f(x)-f(c)}{x-c} \quad \text{or} \quad f'(c)=\lim_{x\to c^-}\frac{f(x)-f(c)}{x-c}.</math>

==Differentiability and continuity==
{{See also|Continuous function}}
[[File:Absolute value.svg|left|thumb|The [[absolute value]] function is continuous (i.e. it has no gaps). It is differentiable everywhere ''except'' at the point {{math|''x''}} = 0, where it makes a sharp turn as it crosses the {{math|''y''}}-axis.]]
[[File:Cusp at (0,0.5).svg|thumb|right|A [[cusp (singularity)|cusp]] on the graph of a continuous function. At zero, the function is continuous but not differentiable.]]
If {{math|''f''}} is differentiable at a point {{math|''x''0}}, then {{math|''f''}} must also be [[continuous function|continuous]] at {{math|''x''0}}. In particular, any differentiable function must be continuous at every point in its domain. ''The converse does not hold'': a continuous function need not be differentiable. For example, a function with a bend, [[cusp (singularity)|cusp]], or [[vertical tangent]] may be continuous, but fails to be differentiable at the location of the anomaly.

Most functions that occur in practice have derivatives at all points or at [[Almost everywhere|almost every]] point. However, a result of [[Stefan Banach]] states that the set of functions that have a derivative at some point is a [[meagre set]] in the space of all continuous functions.<ref>{{cite journal |last=Banach |first=S. |title=Über die Baire'sche Kategorie gewisser Funktionenmengen |journal=[[Studia Mathematica|Studia Math.]] |volume=3 |issue=1 |year=1931 |pages=174–179 |doi=10.4064/sm-3-1-174-179 |doi-access=free }}. Cited by {{cite book|author1=Hewitt, E |author2=Stromberg, K|title=Real and abstract analysis|publisher=Springer-Verlag|year=1963|pages=Theorem 17.8|no-pp=true}}</ref> Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the [[Weierstrass function]].
{{-}}

==Differentiability classes==
[[File:Approximation of cos with linear functions without numbers.svg|300px|thumb|Differentiable functions can be locally approximated by linear functions.]]
[[File:The function x^2*sin(1 over x).svg|thumb|300px|The function <math>f : \R \to \R</math> with <math>f(x) = x^2\sin\left(\tfrac 1x\right)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math> is differentiable. However, this function is not continuously differentiable.]]
{{main|Smoothness}}
A function <math display="inline">f</math> is said to be {{em|{{visible anchor|continuously differentiable|Continuous differentiability}}}} if the derivative <math display="inline">f^{\prime}(x)</math> exists and is itself a continuous function. Although the derivative of a differentiable function never has a [[jump discontinuity]], it is possible for the derivative to have an [[Classification of discontinuities#Essential discontinuity|essential discontinuity]]. For example, the function
<math display="block">f(x) \;=\; \begin{cases} x^2 \sin(1/x) & \text{ if }x \neq 0 \\ 0 & \text{ if } x = 0\end{cases}</math>
is differentiable at 0, since
<math display="block">f'(0) = \lim_{\varepsilon \to 0} \left(\frac{\varepsilon^2\sin(1/\varepsilon)-0}{\varepsilon}\right) = 0</math>
exists. However, for <math>x \neq 0,</math> [[differentiation rules]] imply
<math display="block">f'(x) = 2x\sin(1/x) - \cos(1/x)\;,</math>
which has no limit as <math>x \to 0.</math> Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, [[Darboux's theorem (analysis)|Darboux's theorem]] implies that the derivative of any function satisfies the conclusion of the [[intermediate value theorem]].

Similarly to how [[continuous function]]s are said to be of {{em|class <math>C^0,</math>}} continuously differentiable functions are sometimes said to be of {{em|class <math>C^1</math>}}. A function is of {{em|class <math>C^2</math>}} if the first and [[second derivative]] of the function both exist and are continuous. More generally, a function is said to be of {{em|class <math>C^k</math>}} if the first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> all exist and are continuous. If derivatives <math>f^{(n)}</math> exist for all positive integers <math display="inline">n,</math> the function is [[Smooth function|smooth]] or equivalently, of {{em|class <math>C^{\infty}.</math>}}
{{-}}

==Differentiability in higher dimensions==

A [[function of several real variables]] {{math|'''f''': '''R'''''m'' → '''R'''''n''}} is said to be differentiable at a point {{math|'''x'''0}} if [[there exists]] a [[linear map]] {{math|'''J''': '''R'''''m'' → '''R'''''n''}} such that
:<math>\lim_{\mathbf{h}\to \mathbf{0}} \frac{\|\mathbf{f}(\mathbf{x_0}+\mathbf{h}) - \mathbf{f}(\mathbf{x_0}) - \mathbf{J}\mathbf{(h)}\|_{\mathbf{R}^{n}}}{\| \mathbf{h} \|_{\mathbf{R}^{m}}} = 0.</math>

If a function is differentiable at {{math|'''x'''0}}, then all of the [[partial derivative]]s exist at {{math|'''x'''0}}, and the linear map {{math|'''J'''}} is given by the [[Jacobian matrix]], an ''n'' × ''m'' matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the [[fundamental increment lemma]] found in single-variable calculus.

If all the partial derivatives of a function exist in a [[Neighbourhood (mathematics)|neighborhood]] of a point {{math|'''x'''0}} and are continuous at the point {{math|'''x'''0}}, then the function is differentiable at that point {{math|'''x'''0}}.

However, the existence of the partial derivatives (or even of all the [[directional derivative]]s) does not guarantee that a function is differentiable at a point. For example, the function {{math|''f'': '''R'''2 → '''R'''}} defined by

:<math display="block">f(x,y) = \begin{cases}x & \text{if }y \ne x^2 \\ 0 & \text{if }y = x^2\end{cases}</math>

is not differentiable at {{math|(0, 0)}}, but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function

:<math>f(x,y) = \begin{cases}y^3/(x^2+y^2) & \text{if }(x,y) \ne (0,0) \\ 0 & \text{if }(x,y) = (0,0)\end{cases}</math>

is not differentiable at {{math|(0, 0)}}, but again all of the partial derivatives and directional derivatives exist.

{{See also|Multivariable calculus|Smoothness#Multivariate differentiability classes}}

==Differentiability in complex analysis==
{{main|Holomorphic function}}
In [[complex analysis]], complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing [[complex number]]s. So, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> is said to be differentiable at <math display="inline">x=a</math> when

:<math>f'(a)=\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{f(a+h)-f(a)}{h}.</math>

Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math>, that is complex-differentiable at a point <math display="inline">x=a</math> is automatically differentiable at that point, when viewed as a function <math>f:\mathbb{R}^2\to\mathbb{R}^2</math>. This is because the complex-differentiability implies that

:<math>\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{|f(a+h)-f(a)-f'(a)h|}{|h|}=0.</math>

However, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> can be differentiable as a multi-variable function, while not being complex-differentiable. For example, <math>f(z)=\frac{z+\overline{z}}{2}</math> is differentiable at every point, viewed as the 2-variable [[Real-valued function|real function]] <math>f(x,y)=x</math>, but it is not complex-differentiable at any point because the limit <math display="inline">\lim_{h\to 0}\frac{h+\bar h}{2h}</math> gives different values for different approaches to 0.

Any function that is complex-differentiable in a neighborhood of a point is called [[holomorphic function|holomorphic]] at that point. Such a function is necessarily infinitely differentiable, and in fact [[Analytic function|analytic]].

==Differentiable functions on manifolds==
{{See also|Differentiable manifold#Differentiable functions}}
If ''M'' is a [[differentiable manifold]], a real or complex-valued function ''f'' on ''M'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate chart defined around ''p''. If ''M'' and ''N'' are differentiable manifolds, a function ''f'': ''M'' → ''N'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate charts defined around ''p'' and ''f''(''p'').

==See also==
* [[Generalizations of the derivative]]
* [[Semi-differentiability]]
* [[Differentiable programming]]

==References==
{{reflist}}

{{Differentiable computing}}

[[Category:Multivariable calculus]]
[[Category:Smooth functions]]

Differentiable function - Revision history

131.0.197.244 at 01:18, 9 June 2025