Approximate differentiation: Jarník points

We investigate Jarník's points for a real function f defined in ℝ, i.e. points x for which ${\displaystyle a{p}_{y\to x}\left|\frac{f\left(y\right)-f\left(x\right)}{y-x}\right|=+\infty }$. In 1970, Berman has proved that the set ${\displaystyle J_f}$ of all Jarník's points for a path f of the one-dimensional Brownian motion is the whole ℝ almost surely. We give a simple explicit construction of a continuous function f with ${\displaystyle J_f=ℝ.Thema\in res\underline{t}ofourpapersayst\hat{f}\text{or}aty\pi calcont\in uousfunctionfon[0,1]theset}$J_f!$! is c-dense in [0,1].