I have struggle understanding a certain step in a proof. Let $\boldsymbol{g}:[a,b] \rightarrow \mathbb{R}^n$ be $\mathcal{C}^1$ and $\int_a^b\boldsymbol{g}:=\left[\int_a^bg_1,\, \dots\,,\int_a^bg_n \right]^T$.
At a point in the proof, it is used that
\begin{equation*} \Vert\boldsymbol{g}'(t)\Vert = \lim_{h\rightarrow 0^+}\frac{1}{h}\int_t^{t+h}\Vert\boldsymbol{g}'(u)\Vert du \end{equation*}
Could someone please explain why this is true?