SOLVED
Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continous for every $h>0$. Is it true that
\[f=g+h\]
for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?
A conjecture of Erdős from the early 1950s. Answered in the affirmative by de Bruijn
[dB51].
See also [908].