SCHWEITZER 1978 Problem 4

Let $\mathbb{Q}$ and $\mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $h \in \mathbb{Q} , x_0 \in \mathbb{R}$ , $$f(x+h)-f(x) \rightarrow 0$$ as $x \in \mathbb{Q}$ tends to $x_0$ . Does it follow that $f$ is bounded on some interval?

M. Laczkovich