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Miklós Schweitzer 1961- Problem 8

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December 1, 2015
college contestsreal analysis

Problem Statement

8. Let f(x)f(x) be a convex function defined on the interval [0,12][0, \frac {1}{2}] with f(0)=0f(0)=0 and f(12)=1f(\frac{1}{2})=1; Let further f(x)f(x) be differentiable in (0,12)(0, \frac {1}{2}), and differentiable at 00 and 12\frac{1}{2} from the right and from the left, respectively. Finally, let f′(0)>1f'(0)>1. Extend f(x)f(x) to [0.1][0.1] in the following manner: let f(x)=f(1−x)f(x)= f(1-x) if x∈(12,1]x \in (\frac {1} {2}, 1]. Show that the set of the points xx for shich the terms of the sequence xn+1=f(xn)x_{n+1}=f(x_n) (x0=x;n=0,1,2,…x_0=x; n = 0, 1, 2, \dots ) are not all different is everywhere dense in [0,1][0,1]; (R. 10)
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