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. With a counterexample, we prove that left and right cannot be mixed.