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

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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

Proof of Theorem 25.20 We know from Theorem 17.8 that continuous func- tions with compact support C. (R") are

the restricted maximal function. Since (u-on) 3] 3} } + X"{x:on(x) - u(x) > } Cn "||un||1+0+||on-u||1, where

and, whenever the limit exists and is finite, Du(x) lim- (B, (x)) 0 X" (B(x))* Note that, by Lemma 25.16,

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