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Wikipedia / Differentiable function

In mathematical analysis, a real or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is locally approximable by a linear function at each interior point, and does not contain any break, angle, or cusp.

is said to be differentiable at is said to be continuously differentiable.

Continuous functions may be nowhere differentiable in their domain, such as the Weierstrass function.

For a multivariable function, as shown here, the differentiability of it is something more complex than the existence of the partial derivatives of it. However, the essence is the same; a function that is locally approximable by a linear map is said to be differentiable at this point.

Differentiability of real functions of one variable

, is said to be differentiable if the derivative

exists. This implies that the function is continuous at .

This function is said to be differentiable on if it is differentiable at every point of . In this case, the derivative of is thus a function from

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).

Semi-differentiability

The above definition can be extended to define the derivative at boundary points:

, which is a subset of the reals, it follows that this limit will be defined as either

Differentiability and continuity

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

is said to be continuously differentiable 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 differentiation rules 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 continuously differentiable functions are sometimes said to be of . A function is of if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of the function is smooth or equivalently, of

Differentiability in higher dimensions

A function of several real variables is said to be differentiable at a point if there exists a linear map such that

where for a vector is the norm of it. If a function is differentiable at , then all of the partial derivatives exist at , and the linear map is given by , where , the Jacobian matrix for evaluated at (an n × m matrix in this case), is left-multiplied to a vector in . A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma.

If all the partial derivatives of a function exist in a neighborhood of a point and are continuous at the point , then the function is differentiable at that point ; 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 defined by

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

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

Differentiability in complex analysis

In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed because a complex number when

. This is because complex differentiability implies

as in the definition of the differentiability of higher dimensional spaces above, and the absolute-value norm is expressible as multiplication by a complex number.

corresponds to the 2-variable real function.

A function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such a function is necessarily analytic at that point. Conversely, any function that is analytic at a point is also holomorphic at that point.

Differentiable functions on manifolds

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).