We start with the definition of a continuous function and practice using this definition on various functions. In particular, we prove that polynomials, rational functions, and Cantor functions are continuous in Sect. 4.1. In Sect. 4.2, we introduce the limit of a function in Definition 4.2.1. Some authors define the limit of a function slightly differently as in Definition \(\widehat{4.2.1}\) . So we will explain the distinction between the two by examples in Sect. 4.2. And we explain why we use Definition 4.2.1 rather than Definition \(\widehat{4.2.1}\) . We use Definition 4.2.1 for the limit of a function. I could have omitted Definition \(\widehat{4.2.1}\) , but I thought it offered a good teaching moment to understand the limits of functions. Then we explain the connections between the limit of sequences and functions in order to obtain many theorems from theorems in Chap. 3. For example, the continuity of the exponential function \(a^{x}\) is a consequence of the definition of a number \(a^{r}\) when \(r\) is an irrational number in Chap. 3. In Sect. 4.3, we prove the extreme value theorem and the intermediate value theorem. We show that the inverse function of a continuous function defined on a closed interval is continuous. And we prove that the function \(x^{r}\) is also continuous for any real number \(r\) . In Sect. 4.4, we prove that \(e^{x} = \mathop \sum \limits_{n = 0}^{\infty } \frac{{x^{n} }}{n!}\) . In Sect. 4.5, we briefly introduce a continuous function on a metric space, and mention that further generalization of a metric space to a topological space may be interesting for your further study. We then conclude this chapter by giving an example of a function that is not continuous at rational numbers but continuous at irrational numbers in Sect. 4.6.

Continuous Functions

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发表日期：

2025年

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Hidefumi Katsuura

摘要

We start with the definition of a continuous function and practice using this definition on various functions. In particular, we prove that polynomials, rational functions, and Cantor functions are continuous in Sect. 4.1. In Sect. 4.2, we introduce the limit of a function in Definition 4.2.1. Some authors define the limit of a function slightly differently as in Definition \(\widehat{4.2.1}\) . So we will explain the distinction between the two by examples in Sect. 4.2. And we explain why we use Definition 4.2.1 rather than Definition \(\widehat{4.2.1}\) . We use Definition 4.2.1 for the limit of a function. I could have omitted Definition \(\widehat{4.2.1}\) , but I thought it offered a good teaching moment to understand the limits of functions. Then we explain the connections between the limit of sequences and functions in order to obtain many theorems from theorems in Chap. 3. For example, the continuity of the exponential function \(a^{x}\) is a consequence of the definition of a number \(a^{r}\) when \(r\) is an irrational number in Chap. 3. In Sect. 4.3, we prove the extreme value theorem and the intermediate value theorem. We show that the inverse function of a continuous function defined on a closed interval is continuous. And we prove that the function \(x^{r}\) is also continuous for any real number \(r\) . In Sect. 4.4, we prove that \(e^{x} = \mathop \sum \limits_{n = 0}^{\infty } \frac{{x^{n} }}{n!}\) . In Sect. 4.5, we briefly introduce a continuous function on a metric space, and mention that further generalization of a metric space to a topological space may be interesting for your further study. We then conclude this chapter by giving an example of a function that is not continuous at rational numbers but continuous at irrational numbers in Sect. 4.6.