8. Let f(x) be a convex function defined on the interval [0,21] with f(0)=0 and f(21)=1; Let further f(x) be differentiable in (0,21), and differentiable at 0 and 21 from the right and from the left, respectively. Finally, let f′(0)>1.
Extend f(x) to [0.1] in the following manner: let f(x)=f(1−x) if x∈(21,1].
Show that the set of the points x for shich the terms of the sequence xn+1=f(xn) (x0=x;n=0,1,2,…) are not all different is everywhere dense in [0,1]; (R. 10) college contestsreal analysis