Editing Continuous function (section)

====Weierstrass and Jordan definitions (epsilon–delta) of continuous functions====
[[File:Example of continuous function.svg|right|thumb|Illustration of the {{mvar|ε}}-{{mvar|δ}}-definition: at {{math|1=''x'' = 2}}, any value {{math|δ ≤ 0.5}} satisfies the condition of the definition for {{math|1=''ε'' = 0.5}}.]]
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function <math>f : D \to \mathbb{R}</math> as above and an element <math>x_0</math> of the domain <math>D</math>, <math>f</math> is said to be continuous at the point <math>x_0</math> when the following holds: For any positive real number <math>\varepsilon > 0,</math> however small, there exists some positive real number <math>\delta > 0</math> such that for all <math>x</math> in the domain of <math>f</math> with <math>x_0 - \delta < x < x_0 + \delta,</math> the value of <math>f(x)</math> satisfies
<math display="block">f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.</math>

Alternatively written, continuity of <math>f : D \to \mathbb{R}</math> at <math>x_0 \in D</math> means that for every <math>\varepsilon > 0,</math> there exists a <math>\delta > 0</math> such that for all <math>x \in D</math>:
<math display="block">\left|x - x_0\right| < \delta ~~\text{ implies }~~ |f(x) - f(x_0)| < \varepsilon.</math>

More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small [[Topological neighborhood |neighborhood]] around <math>f\left(x_0\right),</math> we need to choose a small enough neighborhood for the <math>x</math> values around <math>x_0.</math> If we can do that no matter how small the <math>f(x_0)</math> neighborhood is, then <math>f</math> is continuous at <math>x_0.</math>

In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)|basis for the topology]], here the [[metric topology]].

Weierstrass had required that the interval <math>x_0 - \delta < x < x_0 + \delta</math> be entirely within the domain <math>D</math>, but Jordan removed that restriction.