# Differentiable function

(Redirected from Differentiability)

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. Differentiable function_sentence_0

In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. Differentiable function_sentence_1

The differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. Differentiable function_sentence_2

More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. Differentiable function_sentence_3

In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). Differentiable function_sentence_4

The function f is also be called locally linear at x0 as it is well approximated by a linear function near this point. Differentiable function_sentence_5

## Differentiability and continuity Differentiable function_section_1

If f is differentiable at a point x0, then f must also be continuous at x0. Differentiable function_sentence_7

In particular, any differentiable function must be continuous at every point in its domain. Differentiable function_sentence_8

The converse does not hold: a continuous function need not be differentiable. Differentiable function_sentence_9

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. Differentiable function_sentence_10

Most functions that occur in practice have derivatives at all points or at almost every point. Differentiable function_sentence_11

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. Differentiable function_sentence_12

Informally, this means that differentiable functions are very atypical among continuous functions. Differentiable function_sentence_13

The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. Differentiable function_sentence_14

## Differentiability classes Differentiable function_section_2

Main article: Smoothness Differentiable function_sentence_15

A function f is said to be continuously differentiable if the derivative f′(x) exists and is itself a continuous function. Differentiable function_sentence_16

Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. Differentiable function_sentence_17

For example, the function Differentiable function_sentence_18

is differentiable at 0, since Differentiable function_sentence_19

exists. Differentiable function_sentence_20

However, for x ≠ 0, differentiation rules imply Differentiable function_sentence_21

which has no limit as x → 0. Differentiable function_sentence_22

Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Differentiable function_sentence_23

Continuously differentiable functions are sometimes said to be of class C. A function is of class C if the first and second derivative of the function both exist and are continuous. Differentiable function_sentence_24

More generally, a function is said to be of class C if the first k derivatives f′(x), f′′(x), ..., f(x) all exist and are continuous. Differentiable function_sentence_25

If derivatives f exist for all positive integers n, the function is smooth or equivalently, of class C. Differentiable function_sentence_26

## Differentiability in higher dimensions Differentiable function_section_3

A function of several real variables f: R → R is said to be differentiable at a point x0 if there exists a linear map J: R → R such that Differentiable function_sentence_28

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. Differentiable function_sentence_29

A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. Differentiable function_sentence_30

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. Differentiable function_sentence_31

However, the existence of the partial derivatives (or even of all the directional derivatives) does not in general guarantee that a function is differentiable at a point. Differentiable function_sentence_32

For example, the function f: R → R defined by Differentiable function_sentence_33

is not differentiable at (0, 0), but all of the partial derivatives and directional derivatives exist at this point. Differentiable function_sentence_34

For a continuous example, the function Differentiable function_sentence_35

is not differentiable at (0, 0), but again all of the partial derivatives and directional derivatives exist. Differentiable function_sentence_36

## Differentiability in complex analysis Differentiable function_section_4

Main article: Holomorphic function Differentiable function_sentence_37

Any function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Differentiable function_sentence_38

Such a function is necessarily infinitely differentiable, and in fact analytic. Differentiable function_sentence_39