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Theorem 8.9: Let A ≤ R, let ƒ : A → R, and let r E A. The following are equivalent: 1. f is continuous at r. 2. For every open interval (c, d) with ƒ (r) € (c, d), there is an open interval (a, b) with r € (a, b) such that whenever x € (a, b) n A, f(x) € (c,d). 3. For every positive real number e, there is a positive real number & such that whenever x = (r− 8,r + 8) n A, ƒ(x) = (ƒ(r) − €, ƒ (r) + €). 4.0 Vx € A (\x−r] < 8 → [ƒ (x) − ƒ(r)| < €).

Theorem 8.9: Let A ≤ R, let ƒ : A → R, and let r E A. The following are equivalent: 1. f is continuous at r. 2. For every open interval (c, d) with ƒ (r) € (c, d), there is an open interval (a, b) with r € (a, b) such that whenever x € (a, b) n A, f(x) € (c,d). 3. For every positive real number e, there is a positive real number & such that whenever x = (r− 8,r + 8) n A, ƒ(x) = (ƒ(r) − €, ƒ (r) + €). 4.0 Vx € A (\x−r] < 8 → [ƒ (x) − ƒ(r)| < €).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Theorem 8.9:** Let \( A \subseteq \mathbb{R} \), let \( f: A \rightarrow \mathbb{R} \), and let \( r \in A \). The following are equivalent: 1. \( f \) is continuous at \( r \). 2. For every open interval \( (c, d) \) with \( f(r) \in (c, d) \), there is an open interval \( (a, b) \) with \( r \in (a, b) \) such that whenever \( x \in (a, b) \cap A, f(x) \in (c, d) \). 3. For every positive real number \( \epsilon \), there is a positive real number \( \delta \) such that whenever \( x \in (r - \delta, r + \delta) \cap A, f(x) \in (f(r) - \epsilon, f(r) + \epsilon) \). 4. \( \forall \epsilon > 0 \exists \delta > 0 \forall x \in A (|x - r| < \delta \rightarrow |f(x) - f(r)| < \epsilon) \).
Transcribed Image Text:**Theorem 8.9:** Let \( A \subseteq \mathbb{R} \), let \( f: A \rightarrow \mathbb{R} \), and let \( r \in A \). The following are equivalent: 1. \( f \) is continuous at \( r \). 2. For every open interval \( (c, d) \) with \( f(r) \in (c, d) \), there is an open interval \( (a, b) \) with \( r \in (a, b) \) such that whenever \( x \in (a, b) \cap A, f(x) \in (c, d) \). 3. For every positive real number \( \epsilon \), there is a positive real number \( \delta \) such that whenever \( x \in (r - \delta, r + \delta) \cap A, f(x) \in (f(r) - \epsilon, f(r) + \epsilon) \). 4. \( \forall \epsilon > 0 \exists \delta > 0 \forall x \in A (|x - r| < \delta \rightarrow |f(x) - f(r)| < \epsilon) \).
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