Under what conditions the point $z$ is unique in its range of existence? Theorem: Let $f$ be continuous on $[a,b]$. If the range of $f$ contains $[a,b]$, then the equation $f(x)=x$ has at least one solution $z$ in $[a,b]$, i.e., $f(z)=z$.
My question is:
Under what conditions the point $z$ is unique in its range of existence?  

This is a sufficient condition:
$$
|f(x)-f(y|<|x-y|\quad\forall x,y\in[a,b].
$$
