Suppose $f:\mathbb R \to \mathbb R$ is a function satisfying $f(x)-f(y) \leq k \vert x-y \vert$ for some $k \in \mathbb R$ and $\forall x,y \in \mathbb R$. Is $f$ continuous and differentiable?
By taking $f(x)=\sin \vert x\vert$ implies that $f$ satisfies the above inequality with $k=1$, but $f$ is not differentiable. But I'm unable in proving the continuity/discontinuity. Any ideas?
If you want the inequality to hold for a fixed $k$ and all $x, y$, then:
$$ f(x)-f(y) \le k|x-y| $$ and $$ f(y)-f(x) \le k|y-x| = k|x-y| $$ so in fact: $$ |f(x)-f(y)| \le k|x-y| $$ for all $x$, $y$. I.e. you are looking at Lipschitz functions (which are automatically continuous).