# Definition:Function

## Definition

The process which is symbolised by an operation is called a **function**.

The operand(s) of the operation can be considered to be the input(s).

The output of the **function** is whatever the operation is defined as doing with the operand(s).

A **function** is in fact another name for a mapping, but while the latter term is used in the general context of set theory and abstract algebra, the term **function** is generally reserved for mappings between sets of numbers.

## Also known as

When there is a need to distinguish between this and a partial function, a function is sometimes referred to as a **total function**.

When there is a need to distinguish between this and a multifunction, the term **one-valued function** or **uniform function** can be used, but this is rarely seen.

## Also see

## Historical Note

The term **function**, as used in the modern sense, was first used by Gottfried Wilhelm von Leibniz in $1694$.

The notation $\map f x$ itself appears to have originated with Leonhard Paul Euler.

He used it in two particular contexts:

- particular conventional examples like trigonometric function and powers and the like
- $\map y x$ for an arbitrary curve in the plane.

Up until the time of Joseph Fourier, it was accepted that a function was limited to various classes of expression: a polynomial, a finite combination of elementary functions, a power series or a trigonometric series.

Fourier made the claim that a function of arbitrary shape could be represented by a trigonometric series.

It was not until Johann Peter Gustav Lejeune Dirichlet in $1837$ that the modern definition of function was formulated:

*If in any way a definite value of $y$ is determined corresponding to each value of $x$ in a given interval, then $y$ is called a function of $x$.*

The concept of a **mapping** between arbitrary sets which are not necessarily the real or complex numbers arose in the late $19$th century.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1937: Richard Courant:
*Differential and Integral Calculus: Volume $\text { I }$*(2nd ed.) ... (next): Chapter $\text I$: Introduction - 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (next): Chapter $\text I$: Functions and Limits: $\S 2$: Functions - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.1$. Mappings: Example $45$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.1$: Motivation - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 7.1$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Functions