If the slope of a secant is always irrational, is the function linear?

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a function, such that all distinct $x,y \in \mathbb{R}$ we have that:

$$\frac{f(x)-f(y)}{x-y} \not\in \mathbb{Q}$$

Can we conclude that $f$ is a linear function?