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(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x € A, then f(x) = 0 for all x € R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x € A, then f(x) = g(x) for all x € R. Hint. Ilse nart (a

(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x € A, then f(x) = 0 for all x € R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x € A, then f(x) = g(x) for all x € R. Hint. Ilse nart (a

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...
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(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x E A, then f(x) = 0 for all x E R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x E A, then f(x) = g(x) for all x E R. Hint. Use part (a). %3! %3D
Transcribed Image Text:(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x E A, then f(x) = 0 for all x E R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x E A, then f(x) = g(x) for all x E R. Hint. Use part (a). %3! %3D
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