Basic Analysis I — Real Analysis · Problem 5.3.10

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · odd-even-functions-integral

Problem

A function ff is an odd function if f(x)=−f(−x)f(x) = -f(-x), and ff is an even function if f(x)=f(−x)f(x) = f(-x). Let a>0a > 0. Assume ff is continuous. Prove: a) If ff is odd, then ∫−aaf=0\int_{-a}^{a} f = 0. b) If ff is even, then ∫−aaf=2∫0af\int_{-a}^{a} f = 2\int_0^a f.

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