- #1
CarmineCortez
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Homework Statement
Let I be an interval in R, and let f: I-->R be one-to-one, continuous function. Then prove that f^(-1):f(I)-->R is also continuous.
Homework Equations
The Attempt at a Solution
I started a thread yesterday and had some responses but the proofs became quite complicated and my proof ended up wrong, so I tried more basic approach.
So if f is continuous then
limx->af(x) = f(a) now if f-1 is continuous then
limf(x)->f(a)f-1(f(x)) = a
which is equivalent to limf(x)->f(a)x=a
which is true since f is continuous.
Does this make any sense, I'm not really familiar with taking the limit of the domain as the function is changing. Is there any hope for this proof?
Thanks