Continuity of left translation as a function of the group elements

Question

I'm trying to figure out if the following holds.

Let $G$ be a locally compact with Haar measure.

Is the map $x\mapsto L_xf$ continuous as a map from $G$ to $L^p(G)$??

I guess that it is true but I have had a hard time proving it.

I have tried to use the fact that the functions of $C_c(G)$ are left uniformly continuous i.e. $\Vert L_yf-f\Vert_\text{sup}\rightarrow 0$ as $y\rightarrow1$ and afterwards use that these functions are dense in $L^p(G)$, but my problem is that this is with respect to the $p$-norm and not the sup-norm..

Can someone tell me if my statement is true and if so how to prove it??

Answer 1

Consider the case when $G$ is not discrete, ie points have measure 0, then that function is in fact not continuous when we use the supremum-norm.