My attempt:
Because $f(x)$ is continuous at $a$,then $\lim\_{x\rightarrow a^+}f(x)=f(a)$
When $x\rightarrow a^+$,$f(x)$ is strictly decreasing,so for any $x \in (a,b)$, $f(x)>f(a)$. It can also show that $f(x)<f(b)$ in the same way.
Do I prove it in a rigorous way?Thanks in advance!