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ehrenfest
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[SOLVED] Rudin 5.2
Suppose f'(x)>0 in (a,b). Prove that f is strictly increasing in (a,b), and let g be its inverse function. Prove that g is differentiable, and that
g'(f(x))=1/f'(x)
when a<x<b.
I can prove that f is strictly increasing. I cannot prove that the inverse exists. Do I have to go back to the epsilon-delta definition of the derivative for this or is there some clevel way I can manipulate the limits...
Homework Statement
Suppose f'(x)>0 in (a,b). Prove that f is strictly increasing in (a,b), and let g be its inverse function. Prove that g is differentiable, and that
g'(f(x))=1/f'(x)
when a<x<b.
Homework Equations
The Attempt at a Solution
I can prove that f is strictly increasing. I cannot prove that the inverse exists. Do I have to go back to the epsilon-delta definition of the derivative for this or is there some clevel way I can manipulate the limits...