In the definition of contraction mapping, it requires: $(1)$ the function $f$ must map the domain $A$ to $A$, where $A\subseteq\mathbb{R}$. $(2)$ There exists a constant $0<c<1$, such that $\forall x,y\in A, |f(x)-f(y)|\le c|x-y|$ My question is, why it must require $(1)$, and is there a counter-...