Editing Function of a real variable (section)

=== Algebraic structure ===

The arithmetic operations may be applied to the functions in the following way:
* For every real number ''r'', the [[constant function]] <math>(x)\mapsto r</math>, is everywhere defined.
* For every real number ''r'' and every function ''f'', the function <math>rf:(x)\mapsto rf(x)</math> has the same domain as ''f'' (or is everywhere defined if ''r'' = 0).
* If ''f'' and ''g'' are two functions of respective domains ''X'' and ''Y'' such that {{nowrap|''X''∩''Y''}} contains an open subset of <math>\mathbb{R}</math>, then <math>f+g:(x)\mapsto f(x)+g(x)</math> and <math>f\,g:(x)\mapsto f(x)\,g(x)</math> are functions that have a domain containing {{nowrap|''X''∩''Y''}}.

It follows that the functions of ''n'' variables that are everywhere defined and the functions of ''n'' variables that are defined in some [[neighbourhood (mathematics)|neighbourhood]] of a given point both form [[commutative algebra (structure)|commutative algebras]] over the reals (<math>\mathbb{R}</math>-algebras).

One may similarly define <math>1/f:(x)\mapsto 1/f(x),</math> which is a function only if the set of the points {{nowrap|(''x'')}} in the domain of ''f'' such that {{nowrap|''f''(''x'') ≠ 0}} contains an open subset of <math>\mathbb{R}</math>. This constraint implies that the above two algebras are not [[field (mathematics)|fields]].