Basic Analysis I — Real Analysis · Problem 5.2.10

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · properties-of-integral

Problem

Suppose f:[a,b]→Rf : [a,b] \to \mathbb{R} is bounded and has finitely many discontinuities. Show that as a function of xx the expression ∣f(x)∣|f(x)| is bounded with finitely many discontinuities and is thus Riemann integrable. Then show ∣∫abf(x) dx∣≤∫ab∣f(x)∣ dx.\left|\int_a^b f(x)\,dx\right| \leq \int_a^b |f(x)|\,dx.

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