Uniform continuity and translation operators on topological groups

If $G$ is a compact group, $f : G \to \mathbb {F}$ is a continuous function, and $\varepsilon > 0$, show that there is a neighborhood $U$ of the identity in $G$ such that $\left|f(x) - f(y)\right| < \varepsilon$ whenever $xy^{-1} \in U$. (Note that this says that every continuous function on a compact group is uniformly continuous.)