Prove that any onto strictly increasing map $f: (0,1) \to (0,1)$ is continuous.
Since its strictly increasing then for $x<y$ it implies that $f(x) < f(y)$. For continuity I must show that for any $y\in (0,1)$ there exists a $\delta>0$ such that for $\epsilon>0$ then $|x - y|<\delta$ implies that $|f(x) - f(y)|<\epsilon.$