How can I specify a function object without naming it?

Question

I would like a notational way of referencing a particular function object, without naming it anything in particular.

For example, the statement

$f:\mathbb{R}\rightarrow \mathbb{R}, f(x) = x^2$

Makes the name f an identifier for the function object having:

• domain $\mathbb{R}$
• codomain $\mathbb{R}$

and behaving according to a certain function rule. However, is there a notation that will expand to a function object itself, without giving it a name? So that we can then write

$f = \mathrm{somenotation}$

to set $f$ to that function object?

Answer 1

You could try $$ x \in \mathbb R \mapsto x^2 \in \mathbb R$$

Answer 2

It seems to me that $$ f : \mathbb{R}\to\mathbb{R} : x \mapsto x^2 $$ gets the job done. This declares the name of the function ($f$), indicates the domain and codomain of $f$, and specifies a formula by which $f$ acts ($x \mapsto x^2$, i.e. $f(x) = x^2$). If you won't want to give this thing a name, you could probably leave off the $f$ at the beginning, and perhaps write $$ \mathbb{R}\to\mathbb{R} : x \mapsto x^2. $$