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As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are méasurements of the change of output values associated to a change of input values, and these méasurements are the basis of calculus.

The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x3.

During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis).

By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only théoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis has shown that these functions are actually more common than differentiable functions. Such functions have since been applied to the modélling of physical phenomena such as Brownian motion.

Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevcky independently and almost simultanéously gave the modérn "formal" definition of function (see formal definition below).

In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modérn definition and Euler's definition are negligible.

The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic by méans of the lambda calculus.