Let $M \subseteq \mathbb{R}$ be closed and the mapping $T : M \rightarrow M$ fulfills
$$ |T(x)-T(y)| \leq |x-y| $$ $\forall x,y \in M, x \neq y$
Prove or disprove that T has exactly a fixed point.
So here the Lipschitz constant is L=1, we know that if $L \in (0,1)$, then it is a contraction and the mapping has a fixed point, but how can I prove or disprove here that it has exactly one fixed point?