I'm a student taking a real analysis course at university. I'm working down my problem sheet and have been asked the following (which is the full version of the shortened question in the title).
Q:Suppose $f:[0,2 \pi] \rightarrow \mathbb{R}$ is a continuous function satisfying $f(0) = f(2 \pi)$. Show that there is a number $c \in [0,\pi]$ such that $f(c) = f(c+ \pi)$.
The question seems to be referencing $\sin(x)$. A solution I know exists in the range $[0, \pi]$ is $c=0$, therefore $\sin(0)= \sin(\pi)$. However $\sin(x)$ doesn't appear to be directly referenced and I am not sure how to show a value for $c$ only knowing $f(0) = f(2 \pi)$.
If anyone can help it would be much appreciated!
