Uniform continuity of the left translation operator

Question

Let $G$ be a locally compact group. Does the following statement hold?

Let $f\in L^\infty(G)$. The map $x\to L_xf$ is uniformly continuse map from $G$ to $L^\infty(G)$ if and only if $G$ is discrete.

Note that $L_xf(t)=f(x^{-1}t)$.