zmudz
  • zmudz
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.
Mathematics
chestercat
  • chestercat
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Zarkon
  • Zarkon
what have you tried
Zarkon
  • Zarkon
the proof is quite short
zmudz
  • zmudz
I am having trouble even getting started. I don't know what I'm supposed to do. Any help is greatly appreciated.

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Zarkon
  • Zarkon
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

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