Prove the inverse function theorem

[wany]

Homework Statement

Suppose that f' exists and is continuous on a nonempty, open interval (a,b) with $f'(x) \neq 0$ for all $x \in (a,b)$.
I already proved that f is 1-1 on (a,b) and takes (a,b) onto some open interval (c,d).
(i)Show that $f^{-1}$ is continuously differentiable on (c,d).
(ii)Using the function f(x)=x^3 show that (i) is false if the assumption $f'(x) \neq 0$ fails to hold for some $x \in (a,b)$.

Homework Equations

The Attempt at a Solution

Just to make sure to prove that f is 1-1 I did this through:
Proof by contradiction: suppose f is not 1-1 on (a,b)
therefore the slope must change signs or be 0 for some x in (a,b). So f' must be 0 or change signs. Since f' is continuous f' will =0 for some value. But this is a contradiction. So f has to be 1-1.
since f' exists on (a,b), f:(a,b) $\rightarrow$ (c,d).(i)By definition, since f' is continuous on (a,b) then f is continuously differentiable on (a,b).
The Inverse Function Theorem let's us say since f' exists and is continuous on (a,b), that is for any $x \in (a,b)$, f'(x)=y such that $y \in (c,d)$.
Then from this can we say that $f^{-1}$ is differentiable on (c,d) or is another step required, such that f is onto (c,d)?

(ii) so I am not sure how to do this but we must show that at some point the derivative of f=0 implies that the inverse of f will not be continuous.

 

[fzero]

You need to relate $$(f^{-1})'$$ to $$f'$$ and use the properties of $$f'$$.

 

[wany]

So by the inverse function theorem we know that $(f^{-1})'(c)=\frac{1}{f'(a)}$
assuming that c=f(a)