RO-NMO 2010 Problem 124

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$ , and $$I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].$$ Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$ , for any $h\in [0,1]$ . b) the sequence $(a_n)_{n\ge 1}$ , defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$ , is convergent if and only if $f(0)=0$ .

Calin Popescu