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Consider the function: 0 ;if x is rational 1 ;if x is irrational f(x) = a. Use a 8, e argument to prove that f(x) is not continuous at any point x = a, where "a" is a rational number. Hint: You need the fact that any interval around x = a, contains both rational and irrational numbers whether "a" itself is rational or irrational.

Consider the function: 0 ;if x is rational 1 ;if x is irrational f(x) = a. Use a 8, e argument to prove that f(x) is not continuous at any point x = a, where "a" is a rational number. Hint: You need the fact that any interval around x = a, contains both rational and irrational numbers whether "a" itself is rational or irrational.

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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Course: Real Analysis/Mathematical Analysics (Q3)

Consider the function: f(2) = { 0 ; if x is rational 1 ;if x is irrational } a. Use a 8, e argument to prove that f(x) is not continuous at any point x = a, where "a" is a rational number. Hint: You need the fact that any interval around x = a, contains both rational and irrational numbers whether "a" itself is rational or irrational. b. Find an closed set set E CR such that f-1(E) is not closed and hence f (x) is not continuous on R
Transcribed Image Text:Consider the function: f(2) = { 0 ; if x is rational 1 ;if x is irrational } a. Use a 8, e argument to prove that f(x) is not continuous at any point x = a, where "a" is a rational number. Hint: You need the fact that any interval around x = a, contains both rational and irrational numbers whether "a" itself is rational or irrational. b. Find an closed set set E CR such that f-1(E) is not closed and hence f (x) is not continuous on R
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