Abstract
An open interval is a set of reals of the form (a, b) = { x: a < x < b}. As in §1.4, we are allowing \(a = -\infty\) or \(b = \infty\) or both. A compact interval is a set of reals of the form [a, b] = { x: a ≤ x ≤ b}, where a, b are real. The length of [a, b] is b − a. Recall (§1.5) that a sequence subconverges to L if it has a subsequence converging to L.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The choice can be avoided by selecting the leftmost interval at each stage.
- 2.
This uses the axiom of finite choice (Exercise 1.3.24).
- 3.
\(\sup _{A}f\) and \(\inf _{A}f\) are alternative notations for \(\sup f(A)\) and \(\inf f(A)\).
- 4.
g also depends on a.
- 5.
(2.3.2) with x = 1 was used to sum the geometric series in §1.6.
- 6.
This uses the axiom of countable choice.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hijab, O. (2016). Continuity. In: Introduction to Calculus and Classical Analysis. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-28400-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-28400-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28399-9
Online ISBN: 978-3-319-28400-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)