Let $p : [0,1] \to \mathbb{R}$ be a continuous function. Prove the existence of an increasing and continuous function $w : [0,1] \to \mathbb{R}$ such that : $\lim_{x \to 0} w(x) = 0$ and such that :
$$ \forall t, s \in [0,1], \mid p(t)-p(s) \mid \leq w(\mid t-s\mid)$$
I don't know how to find a function $w$ which fullfils thse conditions. I know that I can find a $\delta$ such that if $\mid t-s \mid \leq \delta$ then $\mid p(t)-p(s) \mid \leq \epsilon$. Yet the fact that $w$ is continuous an dincreasing makes it hard.
Thank you !
Since $[0,1]$ is compact and $p$ is a continuous function then for all $\epsilon > 0$ there is $\delta $ such that :
$$\forall x, y, \mid x-y \mid \leq \delta, \mid p(x)-p(y) \mid \leq \epsilon$$
Thus you can just take :
$$w(x) = \sup \{ \mid p(a)-p(b) \mid \mid a, b \in [0,1], \mid a-b \mid \leq x \}$$