Assume that f is cont. and that $f(x+h)=f(x), \forall h >0$ Show f is bounded and that it attains its min and max values.
Here's an attempt. Let $x_{0}$ be an arbitrary point in f's domain. Then f is cont. on $[x_{0},x_{0}+h]$ Since f is cont. on this closed interveral, f attains its min and max by the extreme value theorem. Hence, $m\leq|f(x)|$ and $|f(x)|\leq M$ Since $f(x)=f(x+h)$, I can just replace and hence f is bounded and cont. I feel like there should be more somewhere.