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There exists x_1,x_2 for an integrable function f

Source:

December 8, 2010

functionintegrationcalculusinequalitiesreal analysisreal analysis unsolved

Problem Statement

Let

f:[0,1]\rightarrow\mathbb{R}

be an integrable function such that:

0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.

Show that there exists

x_1\not= x_2, x_1,x_2\in [0,1]

, such that:

\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}