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1. Let ƒ be a continuous function on R such that f(q) = 0 for all rational numbers q E Q. Prove that f(x) = 0 for all r € R. Hint: Let r be an arbitrary irrational number. Recall Q is dense in R. This means there is a sequence of rational number q„ such that qn → r. Now use sequential continuity to show f(r) = 0.

1. Let ƒ be a continuous function on R such that f(q) = 0 for all rational numbers q E Q. Prove that f(x) = 0 for all r € R. Hint: Let r be an arbitrary irrational number. Recall Q is dense in R. This means there is a sequence of rational number q„ such that qn → r. Now use sequential continuity to show f(r) = 0.

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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4. Let f be a continuous function on R such that f(q) = 0 for all rational numbers q € Q. Prove that f(x) = 0 for all r € R. Hint: Let r be an arbitrary irrational number. Recall Q is dense in R. This means there is a sequence of rational number q, such that qn → r. Now use sequential continuity to show f(r) = 0.
Transcribed Image Text:4. Let f be a continuous function on R such that f(q) = 0 for all rational numbers q € Q. Prove that f(x) = 0 for all r € R. Hint: Let r be an arbitrary irrational number. Recall Q is dense in R. This means there is a sequence of rational number q, such that qn → r. Now use sequential continuity to show f(r) = 0.
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