How does function $x_i \rightarrow f(x_1,x_2, ..., x_i, ..., x_n)$ Lipschitz imply existence of partial derivatives?

Question

How does function $x_i \rightarrow f(x_1,x_2, ..., x_i, ..., x_n)$ Lipschitz imply existence of partial derivatives?

Particularly, is there a way to write this in such form that expresses component-wise differentiability?

Answer 1

This question will help you: Lipschitz continuity implies differentiability almost everywhere. This problem is a little bit hard to solve, but there is a reference in this question with the prove.