Problem: Find all functions $f\colon\Bbb R\to \Bbb R$ that satisfy $|f(x)-f(y)| =3|x-y|$ for all $x,y \in \mathbb{R}$.
My Solution: Take two cases.
Case 1: Assume that $f(x)-f(y)=3(x-y)$. Then $f(x)-3x=f(y)-3y$ and therefore $f(x)=3x+c$.
Case 2: Assume that $f(x)-f(y)=3(y-x)$. Then $f(x)+3x=f(y)+3y$ and therefore $f(x)=-3x+c$.
Is there any problem with my solution?I want to mean is it necessary to state that f is continuous , one-one ,monotone? What is wrong with my argument?