Lebl-Basic-Analysis-I · Problem 5.3.3

The Riemann Integral

Mathematics · fundamental-theorem-of-calculus

Problem

Suppose F:[a,b]→RF : [a,b] \to \mathbb{R} is continuous and differentiable on [a,b]∖S[a,b] \setminus S, where SS is a finite set. Suppose there exists an f∈R([a,b])f \in \mathcal{R}([a,b]) such that f(x)=F′(x)f(x) = F'(x) for x∈[a,b]∖Sx \in [a,b] \setminus S. Show that ∫abf=F(b)−F(a)\int_a^b f = F(b) - F(a).

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