Let $f$ be a function on some real interval $[a,b]$. Suppose that $\forall x\in [a,b]$, there exists a positive constant $C$ such that
$$ |f(x)-f(y)| \leq C|x-y| $$
for all $y \in [a,b]$.


Does each $x \in [a,b]$ have a neighborhood $U$ such that


$$ |f(t)-f(s)| \leq C'|t-s| $$


for some $C' > 0$ and $\forall s,t \in U$?



Question about Lipschitz conditions
$f(t) = t\sin(1/t)$ on $[0,1]$.


