Realted with the of "epsilon-delta" definition of continuity, I am looking some example of a continuous function defined on [0,1] such that $|f(t)-f(s)|<\delta/2$ whenever $|t-s|\leq \delta$.
Of coruse, any continuosly differentiable $f$ with $|f'(t)|<1/2$ for each $t\in [0,1]$ satisfies the above condition, but do you know others examples?
Many thanks in advance for your comments.
Define the periodic sawtooth function $$\phi(t):=\left\{\eqalign{|t|\quad &\qquad(-1\leq t\leq 1),\cr \phi(t+2)&\qquad\forall\>t\in{\mathbb R}\ .\cr}\right.$$
Then choose an $N\gg1$ and put $$f(t):={1\over2}\sum_{n=1}^N{1\over 2^n}\>\phi\bigl(2^n t\bigr)\ .$$