Question: For $a<b$, let $f:[a,b]\to \mathbb{R}$ be a function that is continuously differentiable on $[a,b]$ and injective.
(a) Show that $$\int_{a}^{b} f(x) dx+\int_{f(a)}^{f(b)} f^{-1}(y) dy=bf(b)-af(a).$$
(b) Investigate whether the previous equality holds when assuming that the function $f$ is continuous and injective on $[a,b]$ only.
What I did: For part a, I define a function as $F(x)=\int_{a}^{x} f(t) dt+\int_{f(a)}^{f(x)} f^{-1}(y) dy$ and proved that it is equal to $xf(x)-af(a)$ by using FTC and $F(a)=0$. However, I don't even know where I did use that $f$ is "continuously differentiable". So, how can I give a counterexample for part b? (I am just assuming that being continuous is not enough, it is just an interpretation I make by looking the format of the question.)