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Sequence is convergent iff f(0)=0

Source: Romanian MO 2010 Grade 12

August 6, 2012
functioncalculusderivativeintegrationlimitreal analysisreal analysis unsolved

Problem Statement

Let f:[−1,1]→Rf:[-1,1]\to\mathbb{R} be a continuous function having finite derivative at 00, and I(h)=∫−hhf(x) dx, h∈[0,1].I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1]. Prove that a) there exists M>0M>0 such that ∣I(h)−2f(0)h∣≤Mh2|I(h)-2f(0)h|\le Mh^2, for any h∈[0,1]h\in [0,1]. b) the sequence (an)n≥1(a_n)_{n\ge 1}, defined by an=∑k=1nk∣I(1/k)∣a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|, is convergent if and only if f(0)=0f(0)=0.
Calin Popescu
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