Can a continuous, locally Lipschitz and bounded map be approximated by globally Lipschitz functions?

Question

Let $(X,\| {\cdot}\|)$ be a Banach space, and let $C_b(X)$ denote the space of all, bounded, continuous real-valued function on $X$ with the supremum norm $\|{\cdot}\|_{\infty}$.

Suppose that $f\in C_b(X)$ is a locally Lipschitz function, that is, $f|_{B}$ is Lipschitz continuous for every bounded set $B\subset X$. Would it be possible to find a sequence $\{f_n\}_n\subset C_b(X)$ of Lipschitz continuous functions such that $\|f_n-f\|_{\infty}\to 0$?

Answer 1

Consider $f : \mathbb{R}\to \mathbb{R}, x\mapsto \sin(x^2)$. This is bounded, continuous, and locally Lipschitz, but there is no sequence of globally Lipschitz functions that converges uniformly to $f$ on $\mathbb{R}$.