Left and right on locally compact groups

Let $G$ be a locally compact, non-compact group and $f$ a function defined on $G$; we prove that, if $f$ is uniformly continuous with respect to the left (right) structure on $G$ and with a power integrable with respect to the left (right) Haar measure on $G$, then $f$ must vanish at infinity. We prove that left and right cannot be mixed.