Hi. I need to prove that the inverse of a continuous and monotonic function is also continuous. So far I have shown that if we have

$\displaystyle f: (a,b) \rightarrow R $ then for each $\displaystyle y \in f((a,b))$ and $\displaystyle \epsilon >0$ we have that $\displaystyle f(a)<f(x_0 - \epsilon) <y<f(x_0+\epsilon)<f(b)$ (without loss of generality we can assume that f is increasing)

but I don't know what to do next.

Could you help me?