Given a real number $$extract_itex$$alpha$$/extract_itex$$, the Lagrange number of $$extract_itex$$alpha$$/extract_itex$$ is the supremum of all real numbers $$extract_itex$$L>0$$/extract_itex$$ for which the inequality $$extract_itex$$|alpha -p/q|<(L q^2)^-1$$/extract_itex$$ holds for infinitely many rational numbers $$extract_itex$$p/q$$/extract_itex$$. If Lagrange numbers are less than $$extract_itex$$3$$/extract_itex$$, then they characterize some badly approximable real numbers in the context of Diophantine approximations. Moreover, they can be arranged as a set $$extract_itex$${l_{p/q}: p/qin mathbb{Q}cap $$0,1$$ }$$/extract_itex$$ 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 $$extract_itex$$3$$/extract_itex$$.