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Continuous Injective Function on Compact Set of C

  1. Nov 12, 2011 #1
    Prove that the inverse of a continuous injective function f:A -> ℂ on a compact domain A ⊂ ℂ is also continuous.

    So basically because we're in ℂ, A is closed and bounded, and since f is continuous, the range of f is also bounded. Given a z ∈ A, I can pick some arbitrary δ>0 and because f is bounded find some ε>0 whose associated ball centered at f(z) contains some set of points of which a certain subset of these points come from the delta ball around z. I was also at this point thinking of taking the infimum over all the epsilon that work for the given delta and calling it ε'.

    Now since f is injective I know its inverse function is well-defined on f(A), and moreover, if I can prove that: if you're a point inside the ε' ball than you must come from some point inside the delta ball, then that would prove that the inverse of f is continuous, but I can't seem to prove that. I get the feeling that there is some other nice property of ℂ which would allow me to conclude some key piece of information, but I'm not sure what that is, maybe someone could point me in the right direction, thanks.
     
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  3. Nov 12, 2011 #2

    micromass

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    Let's use the sequential definition of compactness. You must prove that if [itex]f(x_n)\rightarrow f(x)[/itex], then [itex]x_n\rightarrow x[/itex], right??

    But [itex](x_n)_n[/itex] is a sequence in A and A is compact. What can you conclude for that sequence??
     
  4. Nov 12, 2011 #3
    I think that hint did the trick, thanks.
     
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