Let $f$: $R\rightarrow R$ be a continuous periodic function with period $T > 0$.
Show: There is an $x \in R$ such that $$f(x) = f(x + T /2).$$
Trivial case: If $f(x)$ is constant, this statement is always true, but this cannot happen as $T>0$, and for a constant function $T=0$, so we are not dealing with a constant function.
This really feels like an intermediate value theorem question, but I am very used to working with concrete functions and wouldn't immediately know how to apply this to an arbitrary function. What do I know about this function?
- $f(x)$ is continuous, useful.
- $f(x)=f(x+T)$ for every single $x$ in the domain.
My intuition tells me it would be useful to consider the values $x=0$ and $x=T$, but to which function would I apply this?
Best Answer
$g(x)=f(x)-f(x+T/2)$, $g(0)=f(0)-f(T/2)$, $g(T/2)=f(T/2)-f(T)=f(T/2)-f(0)$ apply intermediate value to $[0,T/2]$ since $g(0)$ and $g(T/2)$ have opposite signs.