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!

(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.

Quote:

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Originally Posted by georgec5594

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

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