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Shifting endpoints of integral when it´s periodic

Source: Romanian District Olympiad 2002, Grade XII, Problem 4

October 7, 2018
calculusintegrationbreaking integralperiodic functionPeriodicity

Problem Statement

Let be a continuous and periodic function f:R⟶[0,∞) f:\mathbb{R}\longrightarrow [0,\infty ) of period 1. 1. Show:
a) a∈R  ⟹  ∫aa+1f(x)dx=∫01f(x)dx. a\in\mathbb{R}\implies\int_a^{a+1} f(x)dx =\int_0^1 f(x) dx .
b) lim⁡n→∞∫01f(x)f(nx)dx=(∫01f(x)dx)2. \lim_{n\to\infty} \int_0^1 f(x)f(nx) dx=\left( \int_0^1 f(x) dx \right)^2 .
C. Mortici
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