Abstract
This chapter deals with the basic concepts and results in continuity and differentiability of real-valued functions on the real line, together with their various applications. In the second part, we study sequences of functions and their convergence.
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- 1.
We remark that the proposed formula for \(\delta \) originates in trying to estimate \(|f(x)-1|\) for \(x\) close to \(1\), usually achieved by performing some rough work on the algebraic expressions. Quite often, the guess is obtained by working backwards. This method does require some practice; we will touch on it again and again—see, e.g., Example 4.1.3.2—including some of the proposed exercises.
- 2.
Note that the limit in (4.11) can be alternatively written as \({\lim}_{x \rightarrow a}\, \frac{f(x)-f(a)}{x-a}\).
- 3.
The word maximum has been used for points where a bounded above real-valued function attains its supremum (see Definition 333). Sometimes, in order to emphasize this “global” character in contrast with the “local” character of a local maximum , the term global maximum is used for what we called before just a maximum. The same applies to minimum and global minimum.
- 4.
A partition of a set was defined in Sect. 1.1 as a nonempty family of pairwise disjoint subsets whose union is the given set. In some parts of Real Analysis theory, as here and in integration theory, by a partition of an interval it is understood a finite splitting of the interval; for this, a finite number of its points, including the endpoints, is given.
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© 2015 Springer International Publishing Switzerland
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Montesinos, V., Zizler, P., Zizler, V. (2015). Functions. In: An Introduction to Modern Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-12481-0_4
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DOI: https://doi.org/10.1007/978-3-319-12481-0_4
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