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Functions characterized locally by definite integrals

Source: Romanian NO 2011, grade xii, p.2

October 3, 2019
functionreal analysisintegrationcalculus

Problem Statement

Let be a continuous function f:[0,1]⟶(0,∞) f:[0,1]\longrightarrow\left( 0,\infty \right) having the property that, for any natural number n≥2, n\ge 2, there exist n−1 n-1 real numbers 0<t1<t2<⋯<tn−1<1, 0<t_1<t_2<\cdots <t_{n-1}<1, such that ∫0t1f(t)dt=∫t1t2f(t)dt=∫t2t3f(t)dt=⋯=∫tn−2tn−1f(t)dt=∫tn−11f(t)dt. \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt.
Calculate lim⁡n→∞n1f(0)+∑i=1n−11f(ti)+1f(1). \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} .
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