Show that $f(x + 1/n) = f(x)$ for some $x$, given $f(0) = f(1)$

Question

Let $f: [0,1] \to \mathbb{R}$ be continuous and such that $f(0) = f(1)$. For each $n \in \mathbb{N}$, prove that there exists $x \in [0,1]$ such that $x + 1/n \in [0,1]$ and $f(x + 1/n) = f(x)$.

Don't even know where to start.

Answer 1

Denote $$g(x):=f(x+\frac{1}{n})$$ and $$h(x):=g(x)-f(x)$$

Then, we have $$h(0)=g(0)-f(0)=f(\frac{1}{n})-f(0)$$ $$h(\frac{1}{n})=f(\frac{2}{n})-f(\frac{(1}{n})$$ if $n>1$ and so on.

We have to show that $h$ has a root in $[0,1]$. If $h(\frac{k}{n})=0$ for some $kf(\frac{1}{n})>\cdots >f(1)$. But this is a contradiction to $f(0)=f(1)$.

Hence, we have at least once a different sign and can apply the intermediate value theorem because $h$ is continous.