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A 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. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0f(x0)). The function f may also be called locally linear at x0, as it can be well approximated by a linear function near this point.

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Differentiability of real functions of one variableEdit

A function  , defined on an open set  , is said to be differentiable at   if any of the following equivalent conditions is satisfied:

  • The derivative   exists.
  • There exists a real number   such that  . The number  , when it exists, is equal to  .
  • There exists a function   such that   and  .

Differentiability and continuityEdit

 
The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis.
 
A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.

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 meager 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 classesEdit

 
Differentiable functions can be locally approximated by linear functions.
 
The function   with   for   and   is differentiable. However, this function is not continuously differentiable.

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 C1. A function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives f(x), f(x), ..., f (k)(x) all exist and are continuous. If derivatives f (n) exist for all positive integers n, the function is smooth or equivalently, of class C.

Differentiability in higher dimensionsEdit

A function of several real variables f: RmRn is said to be differentiable at a point x0 if there exists a linear map J: RmRn 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: R2R 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 analysisEdit

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   is said to be differentiable at   when

 

Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function  , that is complex-differentiable at a point   is automatically differentiable at that point, when viewed as a function  . This is because the complex-differentiability implies that

 

However, a function   can be differentiable as a multi-variable function, while not being complex-differentiable. For example,   is differentiable at every point, viewed as the 2-variable real function  , but it is not complex-differentiable at any point.

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 manifoldsEdit

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 fM → 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 alsoEdit

ReferencesEdit

  1. ^ Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia. Math. 3 (1): 174–179. . Cited by Hewitt, E; Stromberg, K (1963). Real and abstract analysis. Springer-Verlag. Theorem 17.8.