is sometimes explicitly defined. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted f|Y. In practice, it is often not harmful to identify f and f|Y, and to omit the subscript |Y. Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation. This means that it is not worthy to explicitly define the domain of a function of a real variable. Algebraic structure The arithmetic operations may be applied to the functions in the following way: For every real number r, the constant function , is everywhere defined. For every real number r and every function f, the function 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 contains an open subset of ℝ, then and are functions that have a domain containing . It follows that the functions of n variables that are everywhere defined and the functions of n variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals (ℝ-algebras). One may similarly define which is a function only if the set of the points in the domain of f such that contains an open subset of ℝ. This constraint implies that the above two algebras are not fields. Continuity and limit Until the second part of 19th century, only continuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space. For defining the continuity, it is useful to consider the distance function of ℝ, which is an everywhere defined function of 2 real variables: A function f is continuous at a point which is interior to its domain, if, for every positive real number , there is a positive real number such that for all such that In other words, may be