Carlo Severini

In this field, Severini proved a generalized version of the Weierstrass approximation theorem. Precisely, he extended the original result of Karl Weierstrass to the class of bounded locally integrable functions, which is a class including particular discontinuous functions as members.[7]

Severini proved Egorov's theorem one year earlier than Dmitri Egorov[8] in the paper (Severini 1910), whose main theme is however sequences of orthogonal functions and their properties.[9]

Severini proved an existence theorem for the Cauchy problem for the non linear hyperbolic partial differential equation of first order

${\displaystyle \left\{{\begin{array}{lc}{\frac {\partial u}{\partial x}}=f\left(x,y,u,{\frac {\partial u}{\partial y}}\right)&(x,y)\in \mathbb {R} ^{+}\times [a,b]\\u(0,y)=U(y)&y\in [a,b]\Subset \mathbb {R} \end{array}}\right.,}$

assuming that the Cauchy data ${\displaystyle U}$ (defined in the bounded interval ${\displaystyle [a,b]}$) and that the function ${\displaystyle f}$ has Lipschitz continuous first order partial derivatives,[10] jointly with the obvious requirement that the set ${\displaystyle \scriptstyle \{(x,y,z,p)=(0,y,U(y),U^{\prime }(y));y\in [a,b]\}}$ is contained in the domain of ${\displaystyle f}$.[11]

According to Straneo (1952, p. 99), he worked also on the foundations of the theory of real functions.[12] Severini also left an unpublished and unfinished treatise on the theory of real functions, whose title was planned to be "Fondamenti dell'analisi nel campo reale e i suoi sviluppi".[13]