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• Exercise 3.2.10 in Section 3.2: Let f: R → R and g: R → R be continuous functions. Suppose that f(r) = g(r) for every rational number r. Prove that f(x) = g(x) for every real number .

• Exercise 3.2.10 in Section 3.2: Let f: R → R and g: R → R be continuous functions. Suppose that f(r) = g(r) for every rational number r. Prove that f(x) = g(x) for every real number .

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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Please do exercise 3.2.10 with detailed explanations

• Exercise 3.2.10 in Section 3.2: Let f: R → R and g: R → R be continuous functions. Suppose that f(r) = g(r) for every rational number r. Prove that f(x) = g(x) for every real number î. • Exercise 3.3.3 in Section 3.3: Suppose ƒ: (0, 1) → R is a continuous function such that the limits lim f(x) and x →0 lim f(x) both exist and equal 0. Prove that f attains an absolute minimum or an absolute maximum on the x→1 interval (0, 1). As usual, "or" is inclusive: there might be a minimum but no maximum, a maximum but no minimum, or both a minimum and a maximum. Remarks The point of Exercise 3.2.10 is that a continuous function on R is completely determined by the values on a suitable countable subset of the domain, namely, a countable dense subset. (More generally, a topological space is called separable when there exists a countable dense subset.) In Exercise 3.3.3, you could also write the indicated limits as the one-sided limits lim f(x) and lim f(x). By the x→1 x →0+ author's definition of limit, however, the variable x is automatically restricted to the declared domain of the function, so the decorations + and are optional here.
Transcribed Image Text:• Exercise 3.2.10 in Section 3.2: Let f: R → R and g: R → R be continuous functions. Suppose that f(r) = g(r) for every rational number r. Prove that f(x) = g(x) for every real number î. • Exercise 3.3.3 in Section 3.3: Suppose ƒ: (0, 1) → R is a continuous function such that the limits lim f(x) and x →0 lim f(x) both exist and equal 0. Prove that f attains an absolute minimum or an absolute maximum on the x→1 interval (0, 1). As usual, "or" is inclusive: there might be a minimum but no maximum, a maximum but no minimum, or both a minimum and a maximum. Remarks The point of Exercise 3.2.10 is that a continuous function on R is completely determined by the values on a suitable countable subset of the domain, namely, a countable dense subset. (More generally, a topological space is called separable when there exists a countable dense subset.) In Exercise 3.3.3, you could also write the indicated limits as the one-sided limits lim f(x) and lim f(x). By the x→1 x →0+ author's definition of limit, however, the variable x is automatically restricted to the declared domain of the function, so the decorations + and are optional here.
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