The important difference from calculus is that the inverse function theorem fails.

As an important result, the inverse function theorem has been given numerous proofs.

By the implicit function theorem, is a diffeomorphism on a neighborhood of .

In particular, there are versions of the inverse and implicit function theorems.

The implicit function theorem will provide an answer to this question.

This statement is also known as the inverse function theorem.

The inverse function theorem can be generalized to functions of several variables.

Note that are known and by the implicit function theorem (which is the slope of the line ).

It is the gradient version of the implicit function theorem.

(now the conditions of the implicit function theorem are fulfilled).