quality:correction
Differentiability in higher dimensions

A function of several real variables f: R → R is said to be differentiable at a point x if there exists a linear map J: R → R such that

If a function is differentiable at x, then all of the partial derivatives exist at x, and the linear map J is given by the Jacobian matrix, an n × m matrix in this case. 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 x and are continuous at the point x, then the function is differentiable at that point x.

However, 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 f: R → 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.