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Math Help - Continuity of inverse of a continuous function

  1. #1
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    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!
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  2. #2
    Senior Member Tinyboss's Avatar
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    (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.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by georgec5594 View Post
    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.
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