Let $f:\Bbb R \longrightarrow \Bbb R$ be a function, and $ \lambda \in (\frac {1}{2},1)$.
For all $ x,y \in \Bbb R$ we have
$\lambda |x-f(x)| \leq |x-y| \Longrightarrow |f(x)-f(y)| \leq |x-y|.$
Does any $\mu \geq 1$ exist such that: $|x-f(y)| \leq \mu |x-f(x)|+|x-y|\text{ ?}$