Editing Continuous function (section)

====Definition in terms of neighborhoods====
A [[neighborhood (mathematics)|neighborhood]] of a point ''c'' is a set that contains, at least, all points within some fixed distance of ''c''. Intuitively, a function is continuous at a point ''c'' if the range of ''f'' over the neighborhood of ''c'' shrinks to a single point <math>f(c)</math> as the width of the neighborhood around ''c'' shrinks to zero. More precisely, a function ''f'' is continuous at a point ''c'' of its domain if, for any neighborhood <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>

As neighborhoods are defined in any [[topological space]], this definition of a continuous function applies not only for real functions but also when the domain and the [[codomain]] are [[topological space]]s and is thus the most general definition. It follows that a function is automatically continuous at every [[isolated point]] of its domain. For example, every real-valued function on the integers is continuous.