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

what have you tried

Answer 2

the proof is quite short

Answer 3

I am having trouble even getting started. I don't know what I'm supposed to do. Any help is greatly appreciated.

Answer 4

choose x=0
then f(0)=f(0+r)=f(r) for all r irrational

so f is constant on the irrational numbers

now you need to extend this to the rational numbers

consider x=a-r where a is a rational number