 Solution

Write $a=2\pi m+r$, where $m$ is an integer and $0\le r<2\pi$. The change of variables $x=y+2\pi m$ and the fact that $f$ is a periodic function giveSplit at $2\pi$, then substitute $y=z+2\pi$ in the second piece:ThereforeThis proves translation invariance of an integral over one period without assuming that $f$ is differentiable.