Here is a function with strange continuity properties: 35. 1 if x is the rational number p/q in f (x) = {9 lowest terms 0 ifx is an irrational number (a) Show that f is discontinuous at c if c is rational. Hint: There exist irrational numbers arbitrarily close to c. (b) Show that f is continuous at c if c is irrational. Hint: Let I be the interval {x : |x – c| < 1}. Show that for any Q > 0, I contains at most finitely many fractions p/q with q < Q. Conclude that there is a 8 such that all fractions in {x : |x – c| < 8} have a denominator larger than Q.
Here is a function with strange continuity properties: 35. 1 if x is the rational number p/q in f (x) = {9 lowest terms 0 ifx is an irrational number (a) Show that f is discontinuous at c if c is rational. Hint: There exist irrational numbers arbitrarily close to c. (b) Show that f is continuous at c if c is irrational. Hint: Let I be the interval {x : |x – c| < 1}. Show that for any Q > 0, I contains at most finitely many fractions p/q with q < Q. Conclude that there is a 8 such that all fractions in {x : |x – c| < 8} have a denominator larger than Q.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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