Local Linearity - Differentiability in Higher Dimensions

Differentiability in Higher Dimensions

A function f: Rm → Rn is said to be differentiable at a point x₀ if there exists a linear map J: Rm → Rn 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.