# Functional equation: $f$ is periodic with every irrational numbers as period

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.

It's easy to see any constant function satisfies the original property. But I don't see how to show this is the only solution.

If $y$ is irrational, the set $x = 0$, $r = y$ to get $f(y) = f(0)$.

If $y$ is rational, then set $x = y$, $r = \pi-y$ to get $f(y) = f(\pi)$

But by the first statement, $f(\pi) = f(0)$. Hence, $f$ is constant.

Wrong question.

The general solution of $f(x)=f(x+r)$ is $f(x)=\theta(x)$ , where $\theta(x)$ is an arbitrary periodic functions with period $r$ .

Adding the condition $f:\mathbb{R}\to\mathbb{R}$ the general solution will be modified to $f(x)=\theta(x)$ , where $\theta(x)$ is an arbitrary periodic $\mathbb{R}\to\mathbb{R}$ functions with period $r$ .

For example a particular solution $f(x)=\sin\dfrac{2\pi x}{r}$ , it is a $\mathbb{R}\to\mathbb{R}$ function, a periodic function with period $r$ and satisfies $f(x)=f(x+r)$ .

• $r$ is an arbitrary irrational number; this function is periodic with every irrational period.
– Ian
Aug 3, 2014 at 4:21