Lebesgue, H., Sur l'approximation des fonctions, Bull. Sciences Math., 22 (1898), 278-287. Here is Lebesgue's beautiful proof of the Weierstrass Theorem. It is based on the idea of approximating the single function |x| by polynomials, and the fact that one can uniformly approximate any continuous function on a closed finite interval by continuous piecewise linear approximants. This is Lebesgue's first paper. He obtained his doctorate 4 years later.

Lebesgue, H., Sur la représentation approchée des fonctions, Rend. Circ. Mat. Palermo 26 (1908), 325-328.

Lebesgue, H., Sur les intégrales singulières, Ann. Fac. Sci. Univ. Toulouse 1 (1909), 25-117. The first paper (see page 114) to introduce and use modulus of continuity (see also page 202 of the next paper).