Peter Lejeune Dirichlet
Peter Lejeune Dirichlet helped loosen the concept of function from its older attachment to algebraic expressions like x^2 + 1. A function could instead be understood abstractly as a correspondence: f: A -> B, with f(x) = y.
His formulation was, more precisely, that "to any x there corresponds a single finite y." That sounds close to the modern notion, but Imre Lakatos, following Hermann Hankel, argues that Dirichlet had not fully developed the concept. One sign is that, when discussing piecewise continuous functions, Dirichlet still spoke as though a function has two values at a point of discontinuity.
Source: Peter Gustav Lejeune Dirichlet, "Introduction of the modern concept of function"