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another hard problem

Source: Romania National Olympiad 1996

August 29, 2005
functionintegrationreal analysisreal analysis unsolved

Problem Statement

Suppose that f:[a,b]→R f: [a,b]\rightarrow \mathbb{R} be a monotonic function and for every x1,x2∈[a,b] x_1,x_2\in [a,b] that x1<x2 x_1<x_2 ,there exist c∈(a,b) c\in (a,b) such that ∫x1x2f(x)dx=f(c)(x1−x2) \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) a) Show that f f be the continuous function on interval (a,b) (a,b) b) Suppose that f f is integrable function on interval [a,b] [a,b] but f f isn't a monotonic function then ,is it the result of part a) right?
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