Prove an x exists with f(x) = f(x + T/2)

Question

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!

Answer 1

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.