Suppose a continuous function $f: (0,2) \to \mathbb R $ attains its minimum at $x_0 \in (0,2)$, prove that the function is not injective.
We need to show there are some $a$ and $b$ such that $f(a)=f(b)$. I think we can first notice that since $f(x_0)$ is minimal, for any $x$, we have: $$ f(x_0) \leq f(x)$$
Now what the statement is saying is that there must also be some $x_1$ such that $f(x_0) \geq f(x_1)$. I cannot use the extreme value theorem because we do not have a closed interval here. How could I proceed?