Prove the function $K$ is defined by $K(x) = 4x(1-x)$ maps the interval $[0,1]$ into itself and it is not a contraction. Prove that it has a fixed point. Why does this not contradict the Contraction Mapping Theorem?
I understand how to show that it maps the interval into itself, how to show it has a fixed point, and that it is not a contraction. I am confused about the Contraction Mapping Theorem part.