Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 4 / a

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.

First let $b>0$. Substitute $x=-y$ in the negative half of the integral. Since $g$ is an odd function,The two halves cancel, soFor $b=0$ both limits coincide. For $b<0$, reverse the limits and apply the result with $-b>0$. Thus the identity holds for every real $b$.