This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f \in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $\forall \epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y \in xU$, it follows that $|f(x) - f(y)| < \epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.