(b) A function f: (0,1) → R that is continuous at all a € (0, 1) except at the points an =for n€ N+, at which it is not continuous. In each part of this question, you are asked to give an example of something, and explain whyit is an example. You may use all definitions, lemmas, theorems etc. from the lecture notes.(a) A set ECR with exactly two limit points.

(a) Consider here Noted that here are limit points of .Because here if write the some…

Answered: (b) A function f: (0, 1) → R that is…

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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Question

(b) A function f: (0,1) → R that is continuous at all a € (0, 1) except at the points an =
for n€ N+, at which it is not continuous.

Expert Solution

Step 1: Examples against given statements

(a) Consider here $E equals open curly brackets open parentheses negative 1 close parentheses to the power of n open parentheses 1 plus 1 over n close parentheses space colon n space element of straight natural numbers close curly brackets$

Noted that here $negative 1 space a n d space 1$ are limit points of $E$ .

Because here if write the some elements of E explicitly ,then we have seen that

$E equals open curly brackets 3 over 2 comma 5 over 4 comma 7 over 6 comma......... close curly brackets union open curly brackets negative 2 comma negative 4 over 3 comma negative 6 over 5 comma....... close curly brackets$

for 1st part observed that sequential elements approaches to $1 space$ and for 2nd part approaches to $negative 1$ .

In other words ,for any nbd containg 1 or -1 , we get infinitely many points in the nbd.

But for any other points we must get at least one nbd such that there is no point of E in that nbd except $1 space o r space minus 1$ .