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Math 2.0

Real functions Proposition

Epsilon-delta definition of a limit

Builds on Limit of a function

Statement

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ , let $a$ be a limit point of $D$ , and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ if and only if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ such that $|f(x) - L| < \varepsilon$ for all $x \in D$ with $0 < |x - a| < \delta$ .

Remarks

Each open interval around a point contains an interval $(L - \varepsilon, L + \varepsilon)$ , and each such interval is open. This gives the equivalence with the definition. Many texts take this condition as the definition of the limit. Mathlib states a form for metric spaces as Metric.tendsto_nhds_nhds.

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Limit point of a set definition
Limit of a function definition
Epsilon-delta definition of a limit statement

Further reading

Web https://en.wikipedia.org/wiki/Limit_of_a_function, (ε, δ)-definition of limit

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Tag	`0GBZDAFPX8`
URI	`m2://0GBZDAFPX8`
Kind	statement, proposition
Writing	W0 stub
MSC	26A03
Status	This copy of the site has no build data. Run `m2 status 0GBZDAFPX8`.
Uses	Limit of a function

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