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Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 14T
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(a) Show that for any two real numbers a and b with a < b, there exists a number x ∈ ℝ \ ℚ such that a < x < b. Hint: look for an x of the form x = r√2, with r ∈ ℚ.

(b) Given a segment I ⊂ ℝ and a continuous function f : I → ℝ, show that if f only takes rational values on I, i.e., if f(I) ⊂ ℚ, then the function f is constant.

I attached an image incase the symbols dont show, if able please provide some explanation with the taken steps. Thank you in advance

(d) Show that for any two real numbers a and b with a ≤ b, there exists a number x E R \ Q such that a < x < b. Hint: look for an x of the form x = r√2, with r E Q. (e) Given a segment ICR and a continuous function f : I→ R, show that if f only takes rational values on I, i.e., if f(I) c Q, then the function f is constant.
Transcribed Image Text:(d) Show that for any two real numbers a and b with a ≤ b, there exists a number x E R \ Q such that a < x < b. Hint: look for an x of the form x = r√2, with r E Q. (e) Given a segment ICR and a continuous function f : I→ R, show that if f only takes rational values on I, i.e., if f(I) c Q, then the function f is constant.
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