A k-lipschitz function

Question

Let $f:M\to \mathbb{R}$ a k-lipschitz function, i.e, $\vert f(x)-f(y)\vert\leq kd(x,y)$, for any $x,y\in M$. Show that $f(x)=\displaystyle \inf_{y\in M}[f(y)+kd(x,y)]=\displaystyle\sup_{y\in M}[f(y)-k d(x,y)]$, for all $x\in M$.

Any hint pls!. Regards

Answer 1

$$ |f(x)-f(y)|\leq kd(x,y)\qquad \iff \qquad f(y)-kd(x,y)\leq f(x)\leq f(y)+kd(x,y) $$ and the equality come for $y=x$

Answer 2

Hint: what is $f\left(y\right)+kd\left(x,y\right)$ at $y=x$?