I think I get what Fujisaki is talking about. My definition of right limit is incomplete. I should have demanded, right at the begining of the definition, that $a$ be an adherent point to the set $D \cap ]a,+\infty[$, otherwise we get this problem, since $[b,a] \cap ]a,a+\delta[=\emptyset$.

There's no need to invoke bijectivity, in the proof we just need $\phi$ to be continuously diff, so that $\phi '$ be continuous and $f(\phi(t))\phi'(t)$ be a continuous and hence integrable function. Instead of the example above, I'll use the following integral: $\int^2_1 \frac{1}{\sqrt{e^x-1}}$...