No. of Continuous Functions

Question

Find all continuous functions $f:\mathbf R\rightarrow \mathbf R$ such that $f(x)-f(y)$ is rational for rational $x-y$. Can someone please help me with the solution ? Any Hint will be appreciated.

Answer 1

There are continuum many (i.e., $|\mathbb R|$) such functions. First of all, there are only $|\mathbb R|$ many continuous functions, so this is an upper bound. On the other hand, for any real $r$, $f(x)=x+r$ satisfies the requrements, so there are at least $|\mathbb R|$ many such functions.