I am currently reading through Groups and Symmetries by Yvette Kosmann-Schwarzbach. I am working through a lemma with proof in the section titled One-Parameter subgroups of $GL(n,\mathbb{K})$, and I am having trouble understanding certain steps.
I understand the first few steps until I get to "To prove $f$ is differentiable, it is sufficient to show that there is a real number $a>0$ such that $\int_0^a f(t)dt$ is invertible."
My question is why is this a sufficient condition for differentiability of $f$ and how do the following steps imply invertibility of $\int_0^a f(t)dt$?