Give a non-differentiable fixed-point function

Question

Give an example of a function $\phi:[-1,1]\to[-1,1]$ with unique fixed point $\phi(0)=0$, that is not differentiable at $x=0$ and where the fixed point iteration $\alpha_{j+1}=\phi(\alpha_j)$ for every $\alpha_1\in[-1,1]$ results in $\alpha_3=\phi^2(\alpha_1)=0$. Could somebody help me out? Thanks!

Answer 1

Define $$ \phi(x)= \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \text{ or } x=1 \\ 1 & \mbox{otherwise.} \end{array} \right. $$