Size of Lipschitz constant in locally Lipschitz constant, does it get smaller if the neighbourhood gets smaller?

Question

Let $F:X \to Y$ be a mapping between Banach spaces. Suppose that $F$ is locally Lipschitz. I want to know how the size of the Lipschitz constant depends on the size of the neighbourhood it applies to. Eg. suppose $x_1,x_2$ are in a ball of radius $R$. Then $$|F(x_1)-F(x_2)| \leq C(R)|x_1-x_2|.$$ Does $C(R) \to 0$ if $R \to 0$???

Answer 1

No, $C(R)$ needn't converge to $0$ for $R\to 0$.

For example look at $F\colon \mathbb{R}\to \mathbb{R}, x\mapsto x$.