A devil's staircase from rotations and irrationality measures for Liouville numbers

From Sturmian and Christoffel words we derive a strictly increasing function $\mathrm{\Delta }:[0,\mathrm{\infty })\to \mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of $\mathrm{\Delta }$ distinguishes some irrationality measures of real numbers.