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| [[File:Polynomialdeg3.svg|thumb|right|A differentiable function]]
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| [[File:Absolute value.svg|right|thumb|The [[absolute value]] function is not differentiable at ''x'' = 0.]]
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| [[File:Approximation of cos with linear functions without numbers.svg|400px|thumb|Differentiable functions can be locally approximated by linear functions.]]
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| In [[calculus]] (a branch of [[mathematics]]), a '''differentiable function''' of one [[real number|real]] variable is a function whose [[derivative]] exists at each point in its [[Domain of a function|domain]]. The [[Graph of a function|graph]] of a differentiable function must have a non-vertical [[tangent line]] at each point in its domain. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, [[Cusp (singularity)|cusps]], or any points with a [[vertical tangent]].
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| More generally, if ''x''<sub>0</sub> is a point in the domain of a function ''f'', then ''f'' is said to be '''differentiable at ''x''<sub>0</sub>''' if the derivative ''f''′(''x''<sub>0</sub>) exists. This means that the graph of ''f'' has a non-vertical tangent line at the point (''x''<sub>0</sub>, ''f''(''x''<sub>0</sub>)). The function ''f'' may also be called '''locally linear''' at ''x''<sub>0</sub>, as it can be well approximated by a [[linear function]] near this point.
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| ==Differentiability and continuity==
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| [[File:WeierstrassFunction.svg|thumb|right|The [[Weierstrass function]] is continuous, but is not differentiable at any point.]]
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| If ''f'' is differentiable at a point ''x''<sub>0</sub>, then ''f'' must also be [[continuous function|continuous]] at ''x''<sub>0</sub>. 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, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
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| Most functions which 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 which have a derivative at some point is a [[meager set]] in the space of all continuous functions.<ref>{{cite journal|author=Banach, S.|title=Uber die Baire'sche Kategorie gewisser Funktionenmengen|journal=Studia. Math.|issue=3|year=1931|pages=174–179}}. Cited by {{cite book|author=Hewitt, E and Stromberg, K|title=Real and abstract analysis|publisher=Springer-Verlag|year=1963|pages=Theorem 17.8|nopp=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]].
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| ==Differentiability classes==
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| {{main|Smooth function}}
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| A function ''f'' is said to be '''continuously differentiable''' if the derivative ''f''<nowiki>'</nowiki>(''x'') exists, and is itself a continuous function. Though the derivative of a differentiable function never has a [[jump discontinuity]], it is possible for the derivative to have an essential discontinuity. For example, the function
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| :<math>f(x) \;=\; \begin{cases} x^2\sin (1/x) & \text{if }x \ne 0 \\ 0 & \text{if }x=0\end{cases}</math> | |
| is differentiable at 0, since
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| :<math>f'(0)=\lim_{\epsilon\to0}\left(\frac{\epsilon^2\sin(1/\epsilon)-0}{\epsilon}\right)=0,</math>
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| exists. However, for ''x''≠0,
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| :<math>f'(x)=2x\sin(1/x)-\cos(1/x)</math>
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| which has no limit as ''x'' → 0. Nevertheless, [[Darboux's theorem (analysis)|Darboux's theorem]] implies that the derivative of any function satisfies the conclusion of the [[intermediate value theorem]].
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| Sometimes continuously differentiable functions are said to be of '''class ''C''<sup>1</sup>'''. A function is of '''class ''C''<sup>2</sup>''' if the first and [[second derivative]] of the function both exist and are continuous. More generally, a function is said to be of '''class ''C''<sup>''k''</sup>''' if the first ''k'' derivatives ''f''′(''x''), ''f''″(''x''), ..., ''f''<sup>(''k'')</sup>(''x'') all exist and are continuous. If derivatives f<sup>(n)</sup> exist for all positive integers n, the function is [[smooth function|smooth]] or, equivalently, of '''class ''C''<sup>''∞''</sup>'''.
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| ==Differentiability in higher dimensions==
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| {{See also|Multivariable calculus}}
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| A [[function of several real variables]] {{math|'''f''': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}} is said to be differentiable at a point {{math|'''x<sub>0</sub>'''}} if [[there exists]] a [[linear map]] {{math|'''J''': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}} such that
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| :<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{h} \|} = \mathbf{0}.</math>
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| If a function is differentiable at {{math|'''x<sub>0</sub>'''}}, then all of the [[partial derivative]]s must exist at {{math|'''x<sub>0</sub>'''}}, in which case the linear map {{math|'''J'''}} is given by the [[Jacobian matrix]]. A similar formulation of the higher-dimensional derivative is provided by the [[fundamental increment lemma]] found in single-variable calculus.
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| Note that existence of the partial derivatives (or even all of the [[directional derivative]]s) does not guarantee that a function is differentiable at a point. For example, the function {{math|''f'': '''R'''<sup>2</sup> → '''R'''}} defined by
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| :<math>f(x,y) = \begin{cases}y & \text{if }y \ne x^2 \\ 0 & \text{if }y = x^2\end{cases}</math> | |
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| 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
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| :<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> | |
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| is not differentiable at {{math|(0, 0)}}, but again all of the partial derivatives and directional derivatives exist.
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| It is known that if the partial derivatives of a function all exist and are continuous in a [[Neighbourhood (mathematics)|neighborhood]] of a point, then the function must be differentiable at that point, and is in fact of class ''C''<sup>1</sup>.
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| ==Differentiability in complex analysis==
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| {{main|Holomorphic function}}
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| In [[complex analysis]], any function that is complex-differentiable in a neighborhood of a point is called [[holomorphic function|holomorphic]]. Such a function is necessarily infinitely differentiable, and in fact [[Analytic function|analytic]].
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| ==Differentiable functions on manifolds==
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| {{See also|Differentiable manifold#Differentiable functions}}
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| 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''. More generally, 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'').
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| ==See also==
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| * [[Semi-differentiability]]
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| * [[Generalizations of the derivative]]
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| ==References==
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| {{reflist}}
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| [[Category:Differential calculus]]
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| [[Category:Multivariable calculus]]
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| [[Category:Smooth functions]]
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