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Differentiability of real functions of one variable

A function f : U → R {\displaystyle f:U\to \mathbb {R} } , defined on an open set U ⊂ R {\textstyle U\subset \mathbb {R} } , is said to be differentiable at a ∈ U {\displaystyle a\in U} if the derivative

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}

exists. This implies that the function is continuous at a.

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

A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as is shown below (in the section 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 Differentiability classes).

Differentiability and continuity

See also: Continuous function

If f is differentiable at a point x0, then f must also be continuous at x0. 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.

Most functions that occur in practice have derivatives at all points or at 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.¹ 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

Main article: Smoothness

A function f {\textstyle f} is said to be continuously differentiable if the derivative f ′ ( x ) {\textstyle f^{\prime }(x)} 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 essential discontinuity. For example, the function f ( x ) = { x 2 sin ⁡ ( 1 / x )  if  x ≠ 0 0  if  x = 0 {\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}} is differentiable at 0, since f ′ ( 0 ) = lim ε → 0 ( ε 2 sin ⁡ ( 1 / ε ) − 0 ε ) = 0 {\displaystyle f'(0)=\lim _{\varepsilon \to 0}\left({\frac {\varepsilon ^{2}\sin(1/\varepsilon )-0}{\varepsilon }}\right)=0} exists. However, for x ≠ 0 , {\displaystyle x\neq 0,} differentiation rules imply f ′ ( x ) = 2 x sin ⁡ ( 1 / x ) − cos ⁡ ( 1 / x ) , {\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)\;,} which has no limit as x → 0. {\displaystyle x\to 0.} 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 implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.

Similarly to how continuous functions are said to be of class C 0 , {\displaystyle C^{0},} continuously differentiable functions are sometimes said to be of class C 1 {\displaystyle C^{1}} . A function is of class C 2 {\displaystyle C^{2}} 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 k {\displaystyle C^{k}} if the first k {\displaystyle k} derivatives f ′ ( x ) , f ′ ′ ( x ) , … , f ( k ) ( x ) {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)} all exist and are continuous. If derivatives f ( n ) {\displaystyle f^{(n)}} exist for all positive integers n , {\textstyle n,} the function is smooth or equivalently, of class C ∞ . {\displaystyle C^{\infty }.}

Differentiability in higher dimensions

A function of several real variables f: Rm → Rn is said to be differentiable at a point x0 if there exists a linear map J: Rm → Rn such that

lim h → 0 ‖ f ( x 0 + h ) − f ( x 0 ) − J ( h ) ‖ R n ‖ h ‖ R m = 0. {\displaystyle \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.}

If a function is differentiable at x0, then all of the partial derivatives exist at x0, and the linear map 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 neighborhood of a point x0 and are continuous at the point x0, then the function is differentiable at that point x0.

However, the existence of the partial derivatives (or even of all the directional derivatives) does not guarantee that a function is differentiable at a point. For example, the function f: R2 → R defined by

f ( x , y ) = { x if  y ≠ x 2 0 if  y = x 2 {\displaystyle f(x,y)={\begin{cases}x&{\text{if }}y\neq x^{2}\\0&{\text{if }}y=x^{2}\end{cases}}}

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

f ( x , y ) = { y 3 / ( x 2 + y 2 ) if  ( x , y ) ≠ ( 0 , 0 ) 0 if  ( x , y ) = ( 0 , 0 ) {\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}

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

See also: Multivariable calculus and Smoothness § Multivariate differentiability classes

Differentiability in complex analysis

Main article: 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 numbers. So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when

f ′ ( a ) = lim h → 0 h ∈ C f ( a + h ) − f ( a ) h . {\displaystyle f'(a)=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(a+h)-f(a)}{h}}.}

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

lim h → 0 h ∈ C | f ( a + h ) − f ( a ) − f ′ ( a ) h | | h | = 0. {\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.}

However, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } can be differentiable as a multi-variable function, while not being complex-differentiable. For example, f ( z ) = z + z ¯ 2 {\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}} is differentiable at every point, viewed as the 2-variable real function f ( x , y ) = x {\displaystyle f(x,y)=x} , but it is not complex-differentiable at any point because the limit lim h → 0 h + h ¯ 2 h {\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}} gives different values for different approaches to 0.

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

1. Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia Math. 3 (1): 174–179. doi:10.4064/sm-3-1-174-179.. Cited by Hewitt, E; Stromberg, K (1963). Real and abstract analysis. Springer-Verlag. Theorem 17.8. https://doi.org/10.4064%2Fsm-3-1-174-179 ↩