Does a periodic function on $\mathbb{R}^n$ have fixed point?

Question

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. We assume that $f$ is periodic: for every $x \in \mathbb{R}^n$ and every $a \in \mathbb{Z}^n$ we find $f(x) = f(x + a)$. Is it true that $f$ must have fixed point?

Answer 1

Since it is periodic and continuous, $f$ is bounded: say $|f(x)|\le B$ for all $x$. So $f$ is a continuous function from the closed ball of radius $B$ about $0$ into itself. Now use the Brouwer fixed-point theorem.