Lipschitz function with $0$ derivative a.e. is constant on connected components

Question

I have to find a reference on a book of the following fact:

a Lipschitz continuous function whose derivative is $0$ wherever it exists is constant on connected components of its domain

Does someone know some book where this statement is proved?

Thank You

Answer 1

Lipschitz functions are absolutely continuous, and many books prove the fundamental theorem of calculus for absolutely continuous functions, from which the result follows. A particular reference for this could be Chapter 7 of Rudin's Real and Complex Analysis.

Answer 2

The definition of "absolutely continuous" makes it clear that a Lipschitz function is absolutely continuous. Now the big theorem about absolutely continuous functions says qed.