1. Consider the sets D = {: ne J}, E = DU {0}. (a) Consider f: E→ R which is continuous and f(1) = 3n-1, VnE J. Determine the value of f (0). D = { 1, 1/1/2018 1/23/11/1/1/₁1 ) 3 E = DU {0} 2 1 2 3 (b) Consider g: DR with g() = 2, Vn E J. Is g a continuous function (on D)? (c) Consider h: ER with h() = 2", Vn e J. Is h a continuous function (on E)? 2 1 conv 8

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Question 1:

Consider the sets \( D = \left\{ \frac{1}{n} : n \in J \right\} \), \( E = D \cup \{0\} \).

#### (a) 
Consider \( f : E \to \mathbb{R} \) which is continuous and \( f\left(\frac{1}{n}\right) = \frac{3n-1}{n} \), \(\forall n \in J\). Determine the value of \( f(0) \).

- **Explanation of the Graph**: 
  - The graph displays a set of points plotted on an xy-plane.
  - The x-axis represents values such as \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), converging to 0 from the right.
  - The y-axis shows corresponding \(f(x)\) values: values near \(\frac{5}{2}\), \(\frac{8}{3}\), and \(\frac{11}{4}\), suggesting the function’s behavior as it converges to 3.

#### (b) 
Consider \( g : D \to \mathbb{R} \) with \( g\left(\frac{1}{n}\right) = 2^n \), \(\forall n \in J\). Is \( g \) a continuous function (on \( D \))?

#### (c)
Consider \( h : E \to \mathbb{R} \) with \( h\left(\frac{1}{n}\right) = 2^n \), \(\forall n \in J\). Is \( h \) a continuous function (on \( E \))?
Transcribed Image Text:### Question 1: Consider the sets \( D = \left\{ \frac{1}{n} : n \in J \right\} \), \( E = D \cup \{0\} \). #### (a) Consider \( f : E \to \mathbb{R} \) which is continuous and \( f\left(\frac{1}{n}\right) = \frac{3n-1}{n} \), \(\forall n \in J\). Determine the value of \( f(0) \). - **Explanation of the Graph**: - The graph displays a set of points plotted on an xy-plane. - The x-axis represents values such as \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), converging to 0 from the right. - The y-axis shows corresponding \(f(x)\) values: values near \(\frac{5}{2}\), \(\frac{8}{3}\), and \(\frac{11}{4}\), suggesting the function’s behavior as it converges to 3. #### (b) Consider \( g : D \to \mathbb{R} \) with \( g\left(\frac{1}{n}\right) = 2^n \), \(\forall n \in J\). Is \( g \) a continuous function (on \( D \))? #### (c) Consider \( h : E \to \mathbb{R} \) with \( h\left(\frac{1}{n}\right) = 2^n \), \(\forall n \in J\). Is \( h \) a continuous function (on \( E \))?
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