Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is it true that$$f=g+h+r$$where $g$ is continuous, $h$ is additive (so $h(x+y)=h(x)+h(y)$), and $r(x+h)-r(x)=0$ for every $h$ and almost all (depending on $h$) $x$?