linguistic:paraphrasing, structural:summarization, structural:reformatting
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 {\displaystyle \mathbb {R} }, which is an everywhere defined function of 2 real variables: d (x, y) = | x − y | {\displaystyle d(x,y)=|x-y|}

A function f is continuous at a point a {\displaystyle a} which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that | f (x) − f (a) | < ε {\displaystyle |f(x)-f(a)|<\varepsilon } for all x {\displaystyle x} such that d (x, a) < φ. {\displaystyle d(x,a)<\varphi.} In other words, φ may be chosen small enough for having the image by f of the interval of radius φ centered at a {\displaystyle a} contained in the interval of length 2 ε centered at f (a). {\displaystyle f(a).} A function is continuous if it is continuous at every point of its domain.

The limit of a real-valued function of a real variable is as follows. Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted

if the following condition is satisfied:For every positive real number ε > 0, there is a positive real number δ > 0 such that

for all x in the domain such that

If the limit exists, it is unique. If a is in the interior of the domain, the limit exists if and only if the function is continuous at a. In this case, we have

When a is in the boundary of the domain of f, and if f has a limit at a, the latter formula allows to "extend by continuity" the domain of f to a.