Definition:Function/Historical Note

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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.