# Let RR denoted the set of real numbers. Find all functions f:RR->RR,satisfying abs(f(x) - f(y)) = 2 abs(x-y) for all x,y belongs to RR.?

Dec 28, 2016

$f \left(x\right) = \pm 2 x + {C}_{0}$

#### Explanation:

If $\left\mid f \left(x\right) - f \left(y\right) \right\mid = 2 \left\mid x - y \right\mid$ then $f \left(x\right)$ is Lipschitz continuous. So the function $f \left(x\right)$ is differentiable. Then following,

$\frac{\left\mid f \left(x\right) - f \left(y\right) \right\mid}{\left\mid x - y \right\mid} = 2$ or
$\left\mid \frac{f \left(x\right) - f \left(y\right)}{x - y} \right\mid = 2$ now

${\lim}_{x \to y} \left\mid \frac{f \left(x\right) - f \left(y\right)}{x - y} \right\mid = \left\mid {\lim}_{x \to y} \frac{f \left(x\right) - f \left(y\right)}{x - y} \right\mid = \left\mid f ' \left(y\right) \right\mid = 2$

so

$f \left(x\right) = \pm 2 x + {C}_{0}$