Introduction to Mathematical Analysis I - Second Edition

151 SECTION 2.6 Exercise 2.6.3 . Suppose A and B are compact subsets of R . Then, by Theorem 2.6.5 , A and B are closed and bounded. From Theorem 2.6.2( c) we get that A ∪ B is closed. Moreover, let M A , m A , M B , m B be upper and lower bounds for A and B , respectively. Then M = max { M A , M B } and m = min { m A , m B } are upper and lower bounds for A ∪ B . In particular, A ∪ B is bounded. We have shown that A ∪ B is both closed and bounded. It now follows from Theorem 2.6.5 that A ∪ B is compact. SECTION 3.1 Exercise 3.1.6 . (a) Observe that when x is near 1 / 2, f ( x ) is near 1 / 2 no matter whether x is rational or irrational. We have | f ( x ) − 1 / 2 | = ( | x − 1 / 2 | , if x ∈ Q ; | 1 − x − 1 / 2 | , if x 6∈ Q . Thus, | f ( x ) − 1 / 2 | = | x − 1 / 2 | for all x ∈ R . Given any ε > 0, choose δ = ε . Then | f ( x ) − 1 / 2 | < ε whenever | x − 1 / 2 | < δ . Therefore, lim x → 1 / 2 f ( x ) = 1 / 2. (b) Observe that when x is near 0 and x is rational, f ( x ) is near 0. However, when f is near 0 and x is irrational, f ( x ) is near 1. Thus, the given limit does not exists. We justify this using the sequential criterion for limits (Theorem 3.1.2 ) . By contradiction, assume that lim x → 0 f ( x ) = `, where ` is a real number. Choose a sequence { r n } of rational numbers that converges to 0, and choose also a sequence { s n } of irrational numbers that converges to 0. Then f ( r n ) = r n and f ( s n ) = 1 − s n and, hence, ` = lim n → ∞ f ( r n ) = 0 and ` = lim n → ∞ f ( s n ) = lim n → ∞ ( 1 − s n ) = 1 . This is a contradiction. (c) By a similar method to part (b), we can show that lim x → 1 f ( x ) does not exists. Solving this problem suggests a more general problem as follows. Given two polynomials P and Q , define the function f ( x ) = ( P ( x ) , if x ∈ Q ; Q ( x ) , if x 6∈ Q . If a is a solution of the equation P ( x ) = Q ( x ) , i.e., P ( a ) = Q ( a ) , then the limit lim x → a f ( x ) exists and the limit is this common value. For all other points the limit does not exist.