Are continuous mappings on a compact metric space Lipschitz?

Question

Continuous mapping on a compact metric space is uniformly continuous is a standard result in real analysis. Lipschitz functions are uniformly continuous. Can the aforementioned result be generalized to Lipschitz? i.e. are all continuous functions on compact metric spaces Lipschitz?

Can we require anything more so that all continuous functions are Lipschitz?

Answer 1

First question: No. For example, $f(x) = \sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.

Answer 2

See $\sqrt X $on [0,1] is satisfies given condition but not Lipschitz