# Definition:Function/Historical Note

## Historical Note on Function

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 - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 1$: Introduction - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.19$: Leibniz ($1646$ – $1716$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.24$: Fourier ($1768$ – $1830$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($1805$ – $1859$)