Continuous Injective Function on Compact Set of C

  • Thread starter Poopsilon
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  • #1
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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.
 

Answers and Replies

  • #2
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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??
 
  • #3
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I think that hint did the trick, thanks.
 

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