Epsilon-delta definition of continuity

Statement

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ , and let $a \in D$ . Then $f$ is continuous at $a$ if and only if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(a)| < \varepsilon$ for all $x \in D$ with $|x - a| < \delta$ .

Remarks

Each open interval around a point contains an interval of the form $(c - r, c + r)$ with $r > 0$ . This gives the equivalence. Many texts take this condition as the definition of continuity. Mathlib states a form for metric spaces as Metric.continuousAt_iff.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Epsilon-delta definition of continuity statement

Where this leads

62 pages build on this one.

Tag	`0W8YHHGCJG`
URI	`m2://0W8YHHGCJG`
Kind	statement, proposition
Writing	W0 stub
MSC	26A15
Status	This copy of the site has no build data. Run `m2 status 0W8YHHGCJG`.
Uses	Continuity at a point

Epsilon-delta definition of continuity

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by