Definition:Mapping/Notation/Warning

Notation for Mapping

The notation:

Let $\map f x$ be a mapping (or function) ...

is an abuse of notation.

If $f: S \to T$ is a mapping, then $\map f x \in T$ for all $x \in S$.

Thus $\map f x$ is a mapping if and only if $\Img f$ is a set of mappings.

The point is that, as used here, $\map f x$ is not a mapping, but it is the image of $x$ under $f$.

Hence it is preferable not to talk about:

the function $\cos x$

but instead should say:

the function $\cos$

or:

the function $x \mapsto \cos x$

although for the latter it would be better to also specify the domain and codomain.

This recommendation is not always followed in the literature, for example:

... the number ... $\map f x$ is called the value of the function $f$ at the point $x$. We shall, however, often let $\map f x$ or $y = \map f x$ denote the function $f$; it will always be clear from the context which of the two meanings of $\map f x$ is used.

-- 1970: Arne Broman: Introduction to Partial Differential Equations

$\mathsf{Pr} \infty \mathsf{fWiki}$, indeed, may itself not be rigorous here either.

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term Function of One Independent Variable: Comment $2.36$
• 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Remark $1$