He is known for his contributions to real analysis, partly collected in his book "Fondamenti per la teorica delle funzioni di variabili reali".

In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers.

Much of what Dini here sets forth concerning such topics as continuous and discontinuous functions, the derivative and the conditions for its existence, series, definite integrals, the properties of the incremental ratio, etc., was entirely original with himself and has since come to be regarded everywhere as basal in the real variable theory.Nell'analisi del XX secolo ha avuto innanzitutto ampio sviluppo la teoria delle funzioni di variabili reali (inaugurata nel 1878 da un libro del Dini) in relazione alle operazioni classiche del calcolo.

[4]Dini worked in the field of mathematical analysis during a time when it was begun to be based on rigorous foundations.

[5] He proved the Dini criterion for the convergence of Fourier series and investigated the potential theory and differential geometry of surfaces, based on work by Eugenio Beltrami.