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Darboux property

Source: RMO 2008, Grade 12, Problem 1

April 30, 2008
functionintegrationcalculusderivativelimitreal analysisreal analysis unsolved

Problem Statement

Let a>0 a>0 and f:[0,∞)→[0,a] f: [0,\infty) \to [0,a] be a continuous function on (0,∞) (0,\infty) and having Darboux property on [0,∞) [0,\infty). Prove that if f(0)\equal{}0 and for all nonnegative x x we have xf(x)≥∫0xf(t)dt, xf(x) \geq \int^x_0 f(t) dt , then f f admits primitives on [0,∞) [0,\infty).
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