Continuous Injective Function on Compact Set of C

In summary, the conversation discusses proving that the inverse of a continuous injective function on a compact domain in ℂ is also continuous. The compactness of the domain allows for the boundedness of the range of the function, which helps in proving the continuity of the inverse function. Using the sequential definition of compactness, it is shown that the inverse function is well-defined and continuous.
  • #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.
 
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  • #2
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
I think that hint did the trick, thanks.
 

What is a continuous injective function?

A continuous injective function is a function that preserves the order and topology of the input set in the output set. This means that the function is continuous, meaning small changes in the input result in small changes in the output, and injective, meaning each element of the output set has a unique element in the input set mapped to it.

What is a compact set in C?

A compact set in C is a set that is both closed and bounded. This means that the set contains all of its limit points and that it is contained within a finite distance from any point in the set.

How can you prove that a function is continuous on a compact set in C?

A function can be proven to be continuous on a compact set in C by using the Heine-Cantor theorem, which states that a function is continuous on a compact set if and only if it is continuous at every point in the set.

What is the importance of a continuous injective function on a compact set in C?

A continuous injective function on a compact set in C is important because it allows for a one-to-one correspondence between the input and output sets, meaning that each element in the output set has a unique element in the input set mapped to it. This property is useful in many areas of mathematics, including analysis and topology.

Can a function be both continuous and injective but not be a continuous injective function on a compact set in C?

Yes, a function can be both continuous and injective but not be a continuous injective function on a compact set in C. This is because the function may not preserve the order and topology of the input set in the output set, which is a necessary condition for a continuous injective function on a compact set.

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