Problem #113 ## Continuity: What It Isn’t and What It Is### Theorem 16Suppose \( f \) and \( g \) are both continuous at \( a \). Then \( f + g \) and \( f \cdot g \) are continuous at \( a \).### ProofWe could use the definition of continuity to prove Theorem 16, but Theorem 14 makes our job much easier. For example, to show that \( f + g \) is continuous, consider any sequence \( x_n \) which converges to \( a \). Since \( f \) is continuous at \( a \), then by Theorem 13, \(\lim_{n \to \infty} f(x_n) = f(a)\). Likewise, since \( g \) is continuous at \( a \), then \(\lim_{n \to \infty} g(x_n) = g(a)\). By Theorem 6 of Chapter 8, \(\lim_{n \to \infty} \left( f(x_n) + g(x_n) \right) = \lim_{n \to \infty} f(x_n) + \lim_{n \to \infty} g(x_n)\) = \( f(a) + g(a) = (f+g)(a) \). Thus by Theorem 13, \( f+g \) is continuous at \( a \). The proof that \( f \cdot g \) is continuous at \( a \) is similar.### Problem 113Use Theorem 14 to show that if \( f \) and \( g \) are continuous at \( a \), then \( f \cdot g \) is continuous at \( a \).By employing Theorem 16 a finite number of times, we can see that a finite sum of continuous functions is continuous. That is, if \( f_1, f_2, \ldots, f_n \) are all continuous at \( a \), then \(\sum_{i=1}^{n} f_i\) is continuous at \( a \). But what about an infinite sum? Specifically, suppose \( f_1, f_2, f_3, \ldots \) are all continuous at \( a \). Consider the following argument.Let \( \varepsilon > 0 \). Since \( f_i \) **Theorem 15: Continuity and Sequences**A function \( f \) is continuous at a point \( a \) if and only if it satisfies the following property:For all sequences \( (x_n) \), if \( \lim_{n \to \infty} x_n = a \), then \( \lim_{n \to \infty} f(x_n) = f(a) \).**Explanation:**Theorem 15 states that to demonstrate the continuity of \( f \) at a point, any sequence \( (x_n) \) converging to \( a \) must result in \( (f(x_n)) \) converging to \( f(a) \). The associated diagram illustrates this idea.**Diagram Description:**The graph shows:- An increasing sequence of points \( a_1, a_2, a_3, a_4, \ldots \) approaching \( a \).- Corresponding function values \( f(a_1), f(a_2), f(a_3), f(a_4), \ldots \) approaching \( f(a) \).**Example 12: Proving Discontinuity**Use Theorem 15 to prove that:\[ f(x) = \begin{cases} 1/x, & x \neq 0 \\ 0, & x = 0 \end{cases}\]is not continuous at 0.**Proof:**Rewrite \( f(x) \) as:\[ f(x) = \begin{cases} 1, & x > 0 \\ -1, & x < 0 \\ 0, & x = 0 \end{cases}\]Select a sequence \( (x_n) = (1/n) \) converging to 0, but \( (f(x_n)) \) does not converge to \( f(0) \).- \(\lim_{n \to \infty} x_n = 0\)- \(\lim_{n \to \infty} f(x_n) = 1 \neq f(0) = 0\)Thus, \( f \) is not continuous at 0 by Theorem 15.**Problems:**- **Problem 109:** Prove that:\[ f(x) = \begin{cases} 1, & x \neq a \\

Answered: Problem 113. Use Theorem to show that…

Problem 113. Use Theorem to show that if / and g are continuous at a, then /g is continnous at a.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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