Prove an x exists with f(x) = f(x + T/2) Suppose $f: \mathbb{R} → \mathbb{R}$ is a continuous and periodic function with period $T > 0$. Show that: there exists an $x \in \mathbb{R}$ such that $f(x) = f(x + T/2)$.
We figured out we needed to use the intermediate value theorem (our instructor approved of the idea), but still can't figure out how to prove it. Any help is appreciated!

As "always" in similar problems, consider $g(x)=f(x)-f(x+T/2)$. Then $g$ is continuous and $g(0)=f(0)-f(T/2)$ and $g(T/2)=f(T/2)-f(T)=f(T/2)-f(0)=-g(0)$. So either $g(0)=0$ already or $g(0)$ and $g(T/2)$ have opposite signs.
