Differentiable Function - Differentiability in Higher Dimensions

Differentiability in Higher Dimensions

See also: Multivariable calculus

A function 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 must exist at x0, in which case 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.

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 ƒ: R2 → 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.

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 is in fact of class C1.

Read more about this topic:  Differentiable Function

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