Show that an irrationally periodic function is also a constant function

Question

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.

Answer 1

Hint: It is enough to show that you have $f(x) = f(x+r)$ for every real number $r$. You have it for all irrational $r$, so you just need to prove it for rational $r$. You can write any rational number as a sum of two irrational numbers.

Answer 2

Given any irrational number r, and rational number k, since k=r+(k−r), the sum of two irrational numbers, for any rational k, f(x)=f(x+k). As this property applies to all real numbers, rational or not, the function must be equal for all real numbers, and henceforth, it is constant.