Undergraduate Mathematics/Differentiable function

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A differentiable function
The absolute value function is not differentiable at .
Differentiable functions can be locally approximated by linear functions.

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

Differentiability and continuity[edit]

The Weierstrass function is continuous, but is not differentiable at any point.

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 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 classes[edit]

A function is said to be continuously differentiable if the derivative 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, since

exists. However, for ,

which has no limit as . Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.

Sometimes continuously differentiable functions are said to be of class . A function is of class if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class if the first derivatives all exist and are continuous. If derivatives exist for all positive integers , the function is smooth or equivalently, of class .

Differentiability in higher dimensions[edit]

See also: Multivariable calculus

A function of several real variables 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 derivatives must exist at , in which case the linear map 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.

Note that existence of the partial derivatives (or even all of the directional derivatives) 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 of a point, then the function must be differentiable at that point, and it is in fact of class .

Differentiability in complex analysis[edit]

In complex analysis, any function that is complex-differentiable in a neighborhood of a point is called holomorphic. Such a function is necessarily infinitely differentiable, and in fact analytic.

Differentiable functions on manifolds[edit]

See also: Differentiable manifold#Differentiable functions

If is a differentiable manifold, a real or complex-valued function on is said to be differentiable at a point if it is differentiable with respect to some (or any) coordinate chart defined around . More generally, if and are differentiable manifolds, a function is said to be differentiable at a point if it is differentiable with respect to some (or any) coordinate charts defined around and .

References[edit]

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