Here is a function with strange continuity properties: 1 if x is the rational number p/q in lowest terms f (x) = if x 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: 1 if x is the rational number p/q in lowest terms f (x) = if x 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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