Criteria for Lipschitz continuity

Question

Is the following statement true. Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup_t\limsup_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}<\infty,$$ then $f$ is Lipchitz continuous.

Answer 1

Note that the assumption of continuity is redundant with the displayed one. Indeed, for each $t\in [0,1]$, we can find $\delta(t)>0$ such that $|f(t)-f(s)|\leq (M+1)|t-s|$ if $|t-s|<\delta$, where $M$ is the supremum involved in the hypothesis. Otherwise, we would be able to find $t\in[0,1]$ and a sequence $\{t_n\}$ converging to $t$ such that $|f(t)-f(t_n)|>(M+1)|t-t_n|$, contradicting the assumption.

The problem has been answered at math.overflow by Misha.