Later on, Joeph Louis Lagrange
(1736-1813) published his manuscripts “Theorie des functions analytiques” in
1793, where he discussed  about analytic function and used the notion f (x), F(x),
φ(x) etc for different function of x Subsequently, Lejeunne Dirichlet
(1805-1859) gave the definition of function which was being used till the set
theoretic definition of function presently used, was given after set theory was
developed by Georg Cantor (1845-1918) The set theoretic definition of function
known to us presently is simply an abstraction of the definition given by Dirichlet
in a rigorous manner