Abstract
The principal aim of this chapter is to give a short introduction to the limit inferior and limit superior and (thereby) the limit for functions. Many (arithmetic and analytic) properties of these functional limits can be derived by establishing their link with sequential limits. In our largely classical approach, continuity and differentiability of real functions are also introduced and treated here as special limits (stopping short of fully developed advanced differential calculus) mainly because the derivative as a limit is an indispensable tool for later developments. For future purposes, we also give quick proofs of the Extreme Values Theorem, the Intermediate Value Theorem, and the Fermat Principle.
“Nothing takes place in the world whose meaning is not that of some maximum or minimum.
Leonhard Euler (1707–1783)
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Notes
- 1.
This property is termed sequential continuity. In our case of single-variable (and also multivariate) real functions, this is equivalent to continuity.
- 2.
By Proposition 4.2.1 since constant functions (such as − 1) are obviously continuous.
- 3.
Since g(c) ≠ 0, we also have g(x) ≠ 0 for |x − c| < δ with \(0<\delta \in \mathbb {R}\) small enough.
- 4.
This statement also holds with [0,  1] replaced by an arbitrary closed interval [a, b]. In this form, it is often termed as the 1-dimensional Brouwer fixed point theorem, even though the latter (in dimensions ≥ 2) is more subtle.
- 5.
Clearly, limx→0 x 0 = 1. This is one of the reasons why sometimes 00 is defined as 1.
- 6.
Algebraic functions will be treated in detail in Section 9.
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Toth, G. (2021). Limits of Real Functions. In: Elements of Mathematics. Undergraduate Texts in Mathematics(). Springer, Cham. https://doi.org/10.1007/978-3-030-75051-0_4
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DOI: https://doi.org/10.1007/978-3-030-75051-0_4
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