If f is continuous on $[a,b]$ and $f(a)=f(b)$ then show that there exists $x,y \in (a,b)$ such that $f(x)=f(y)$
It looks obvious if I imagine the graph. But I am not able to prove it. I am trying to employ intermediate value property, but not able to reach to the conclusion.
Hint: Either $f$ is a constant (so you are done), or there is some $t\in(a,b)$ such that $f(t)\neq f(a)$, in which case employ IVT for $f$ on $[a,t]$ and $[t,b]$.
Without using IVP: from the statement it's clear that $f$ cannot be strictly monotone in $(a,b)$. If for all $x,y \in (a,b)$ we have $x\neq y \Rightarrow f(x) \neq f(y)$ (otherwise we are done), then that tells $f$ is injective and continuous in $(a,b)$. This link says that a continuous and injective function is strictly monotone, raising a contradiction.
