Given a real number $alpha$, the Lagrange number of $alpha$ is the supremum of all real numbers $L>0$ for which the inequality $|alpha -p/q|<(L q^2)^-1$ holds for infinitely many rational numbers $p/q$. If Lagrange numbers are less than $3$, then they characterize some badly approximable real numbers in the context of Diophantine approximations. Moreover, they can be arranged as a set ${l_{p/q}: p/qin mathbb{Q}cap [0,1] }$ using the Farey index. The present talk considers a singular function devised from Sturmian words. After investigating its regularity and singularity, we demonstrate that this function contains all information on Lagrange numbers less than $3$.