Let (u:[a, b] ightarrow mathbb{R}) be a continuous function. Show that (x mapsto int_{[a, x]} u(t) d

Question:

Let $u:[a, b] ightarrow \mathbb{R}$ be a continuous function. Show that $x \mapsto \int_{[a, x]} u(t) d t$ is everywhere differentiable and find its derivative. What happens if we assume only that $u \in \mathcal{L}^{1}(d t)$ ? [ consider Theorem 25.20.]

Data from theorem 25.20

$Theorem 25.20 (Lebesgue) Let u L (X"). The limit 1 lim X" (B,(x)) , \u(v) u(x)[X" (dy) =0 B.(x) exists for$