# Math Help - Continuity of inverse of a continuous function

1. ## Continuity of inverse of a continuous function

I was wondering if someone could help me with a proof:

Suppose we have the function f such that f is:
(i) continuous
(ii) strictly monotonic
(iii) injective

Prove the inverse of f is continuous.

Much appreciated!

2. (Assuming you have already shown or aren't asked to show that f has an inverse in the first place)

It will be enough to show that f is open, that is, for any open set U, f(U) is also an open set. In fact, it suffices to consider open intervals. See where you can go from there.

3. Originally Posted by georgec5594
Suppose we have the function f such that f is:
(i) continuous
(ii) strictly monotonic
(iii) injective

Hypothesis (iii) is irrelevant. If f is strictly monotonic then, f is injective.