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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.

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It's really logic:

Contraction mapping principle: If $f : [0,1] \to [0,1]$ is a contraction, then it has a fixed point.

Your example just showed the converse of the above statement:

If $f : [0,1] \to [0,1]$ has a fixed point, then it is a contraction.

is not true. As the converse of a statement is not the same as that statement, your example does not contradict the contraction mapping principle.

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