Henri Lebesgue (1875-1941)
- 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 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).