I'm having difficults to write formally the first two items could someone show me a formal proof of both. about the last item i have no idea how to start it. I don't know the concept of continuity either derivability so doesn't make sense for me using they to proof it, any suggestion? Question bellow:
Consider $f:[p,q] \to \mathbb{R}$ a strictly increasing function whose image is the interval $[\gamma , \theta]$.
$i)$ Prove that $\gamma = f(p) \ ,\ \theta = f (q)$ and that $f$ is a bijective fuction from $[p,q]$ to $[\gamma, \theta]$.
$ii)$ Prove that $f^{-1}$ is strictly increasing.
$iii)$ For $f(p) \ge 0$ and $f(p) \lt 0$ prove that:
$$\int_{\gamma}^{\theta} f^{-1}(x) dx\ = \ q\ f(q) - p\ f(p) - \int_{p}^{q} f(x) dx $$
Hints:
(i) and (ii) are straightforward and depend only on understanding the words "image" and "inceasing". How could $f(p)$ be smaller than $\gamma$?
For (iii) draw a the graph of $f$ and put it in a rectangle.Think about how the graph of $f$ is related to the graph of its inverse.