In calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

 (a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

), a differentiable function is a function whose derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 exists at each point in its domain. The graph

Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 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, or cusps

Cusp (singularity)

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....

, or any points with a vertical tangent

Vertical tangent

In mathematics, a vertical tangent is tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.- Limit definition :...

.

More generally, if x₀ is a point in the domain of a function ƒ, then ƒ is said to be differentiable at x₀ if the derivative ƒ′(x₀) is defined. This means that the graph of ƒ has a non-vertical tangent line at the point (x₀, ƒ(x₀)), and therefore cannot have a break, bend, or cusp at this point.

Differentiability and continuity

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

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 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 which occur in practice have derivatives at all points or at almost every

Almost everywhere

In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...

 point. However, a result of Stefan Banach

Stefan Banach

Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....

 states that the set of functions which have a derivative at some point is a meager 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

Weierstrass function

In mathematics, the Weierstrass function is a pathological example of a real-valued function on the real line. The function has the property that it is continuous everywhere but differentiable nowhere...

.

Differentiability classes

A function ƒ is said to be continuously differentiable if the derivative ƒ′(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
is differentiable at 0 (with the derivative being 0), but the derivative is not continuous at this point.

Sometimes continuously differentiable functions are said to be of class C¹. A function is of class C² if the first and second derivative

Second derivative

In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...

 of the function both exist and are continuous. More generally, a function is said to be of class C^k if the first k derivatives ƒ′(x), ƒ″(x), ..., ƒ^(k)(x) all exist and are continuous. If derivatives f⁽ⁿ⁾ exist for all positive integers n, the function is smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

 or, equivalently, of class C^∞.

Differentiability in higher dimensions

A function is said to be differentiable at a point if there exists a linear map  such that
If a function is differentiable at , then all of the partial derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s must exist at , in which case the linear map is given by the Jacobian matrix.

Note that existence of the partial derivatives (or even all of the directional derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

s) does not guarantee that a function is differentiable at a point. For example, the function defined by


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


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

It is known that if the partial derivatives of a function all exist and are continuous in a neighborhood

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...

 of a point, then the function must be differentiable at that point, and is in fact of class C¹.

Differentiability in complex analysis

In complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, any function that is complex-differentiable in a neighborhood of a point is called holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

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

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

.

Differentiable functions on manifolds

If M is a differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

, a real or complex-valued function ƒ 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 ƒ: 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 ƒ(p).