Please do it step by step. I will give feedback soon! (Rudin, Ch. 6, Exercise 6) Divide the unit interval [0, 1] into thirds and remove the open intervalG,). What remains is a pair of closed intervals, each of them one-third as long as the original. Foreach of the remaining intervals, do this again, i.e., divide the interval into thirds and remove the openmiddle interval. Repeating the process over and over again, what is left in the limit is the Cantormiddle-thirds set C.Let f : [0, 1]] → R be a bounded function that is continuous at every point in [0, 1] \ C. Show that fis Riemann integrable on [0, 1]. (Hint: Use the argument of Theorem 6.10 from Rudin).

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Description: Please do it step by step. I will give feedback soon! (Rudin, Ch. 6, Exercise 6) Divide the unit interval [0, 1] into thirds and remove the open intervalG,). What remains is a pair of closed intervals, each of them one-third as long as the original.

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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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Please do it step by step. I will give feedback soon!

(Rudin, Ch. 6, Exercise 6) Divide the unit interval [0, 1] into thirds and remove the open interval
G,). What remains is a pair of closed intervals, each of them one-third as long as the original. For
each of the remaining intervals, do this again, i.e., divide the interval into thirds and remove the open
middle interval. Repeating the process over and over again, what is left in the limit is the Cantor
middle-thirds set C.
Let f : [0, 1]] → R be a bounded function that is continuous at every point in [0, 1] \ C. Show that f
is Riemann integrable on [0, 1]. (Hint: Use the argument of Theorem 6.10 from Rudin).

Expert Solution

Step 1

Given that:

The interval [0,1] is given. divide the [0, 1] into 3 subintervals and delete the open middle subinterval and delete the open middle subinterval (1/3, 2/3), leaving two intervals [0,1/3] U [2/3, 1]

Now, again divide each of the 2 resulting intervals above into the open middle third subinterval of each subinterval of each interval obtained in the previous step.

And continue the process, and each step, delete the open middle third subinterval of each interval.

Here objective is to find the limit of the Cantor middle-thirds set C.

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