Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively).
If $x\mapsto f(x-h)-f(x)$ is continuous for a.e. h, then I expect that $f$ is a.e. equal to a continuous function. Is that true?