Example of a discontinous pseudo contraction mapping

Question

$T: X\to X$ is a mapping with a fixed point $x^*$ with a property

$\|T(x)-x^*\|\le \alpha\|x-x^*\|\forall x\in X,\alpha\in[0,1)$, could anyone give an example of such a map but discontinous? or ingeneral how one can proof such map need not be continous?

I am getting an example but that is continous, like

$T:[0,2]\to [0,2], T(x)=\max\{0,x-1\}$

Thanks for helping.

Answer 1

Let $T:\mathbb R\to \mathbb R$ defined by

$$T(x)=\left\{\begin{array}[ll]\ x/2 & x \in\mathbb Q \\ -x/2 & x\notin\mathbb Q\end{array}\right.$$

This is not continuous, has $0$ as a fixed point and $||T(x)-T(0)||=||T(x)-0||=||T(x)||=||x||/2=||x-0||/2$.

So $x^*=0$ and $\alpha=1/2$.

Answer 2

Let $f:\mathbb R \to \mathbb R$ be defined by $f(x)=\frac x 2$ if $|x| \leq 1$ and $f(x)=\frac x 3$ if $|x| > 1$. Here $x^{*}=0$. $f$ is not continuous at $1$ and $-1$.