Prove that $f$ gets its maximum

Question

Let $f$ be a continous positive function in $(0,1)$ and $\lim_{x\to 0^+}(f(x))=\lim_{x\to 1^-}(f(x))=0$ Prove that $f$ gets its maximum in (0,1)

Answer 1

Hint: $f$ extends to a continuous function on $[0,1]$, and you know $\sup f\neq 0$

Answer 2

As noted by user10354138 $f$ can be extended to a continuos function $\overline{f}$ on $[0,1]$, where

$\overline{f}(0)=\overline{f}(1)=0$ (why?).

$ \overline{f}$ attains its maximum on $[0,1]$.

Since $\overline{f}(x)= f(x) >0$, $x \in (0,1)$, the maximum is attained in $(0,1)$.