# Differentiable function

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.

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

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.

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.

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

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

## Differentiability of real functions of one variable

## 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.

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 is said to be continuously differentiable if the derivative f′(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

is differentiable at 0, since

exists.

However, for x ≠ 0, differentiation rules imply

which has no limit as x → 0.

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

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.

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.

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

## Differentiability in higher dimensions

See also: Multivariable calculus

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

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.

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 in general guarantee that a function is differentiable at a point.

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

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

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

## Differentiability in complex analysis

Main article: Holomorphic function

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. 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).

## See also

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Differentiable function.