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

f: (a,b) \rightarrow R then for each y \in f((a,b)) and \epsilon >0 we have that 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?