Introduction to Mathematical Analysis I - Second Edition

77 (a) Let g , h : [ 0 , 1 ] → R be continuous functions and define f ( x ) = ( g ( x ) , if x ∈ Q ∩ [ 0 , 1 ] ; h ( x ) , if x ∈ Q c ∩ [ 0 , 1 ] . Prove that if g ( a ) = h ( a ) , for some a ∈ [ 0 , 1 ] , then f is continuous at a . (b) Let f : [ 0 , 1 ] → R be the function given by f ( x ) = ( x , if x ∈ Q ∩ [ 0 , 1 ] ; 1 − x , if x ∈ Q c ∩ [ 0 , 1 ] . Find all the points on [ 0 , 1 ] at which the function is continuous. 3.3.9 B Consider the Thomae function defined on ( 0 , 1 ] by f ( x ) =   1 q , if x = p q , p , q ∈ N , where p and q have no common factors; 0 , if x is irrational . (a) Prove that for every ε > 0, the set A ε = { x ∈ ( 0 , 1 ] : f ( x ) ≥ ε } is finite. (b) Prove that f is continuous at every irrational point, and discontinuous at every rational point. 3.3.10 B Consider k distinct points x 1 , x 2 , . . . , x k ∈ R , k ≥ 1. Find a function defined on R that is continuous at each x i , i = 1 , . . . , k , and discontinuous at all other points. 3.3.11 Suppose that f , g are continuous functions on R and f ( x ) = g ( x ) for all x ∈ Q . Prove that f ( x ) = g ( x ) for all x ∈ R . 3.4 PROPERTIES OF CONTINUOUS FUNCTIONS Recall from Definition 2.6.3 that a subset A of R is compact if and only if every sequence { a n } in A has a subsequence { a n k } that converges to a point a ∈ A . Theorem 3.4.1 Let D be a nonempty compact subset of R and let f : D → R be a continuous function. Then f ( D ) is a compact subset of R . In particular, f ( D ) is closed and bounded. Proof: Take any sequence { y n } in f ( D ) . Then for each n , there exists a n ∈ D such that y n = f ( a n ) . Since D is compact, there exists a subsequence { a n k } of { a n } and a point a ∈ D such that lim k → ∞ a n k = a ∈ D . It now follows from Theorem 3.3.3 that lim k → ∞ y n k = lim k → ∞ f ( a n k ) = f ( a ) ∈ f ( D ) . Therefore, f ( D ) is compact. The final conclusion follows from Theorem 2.6.5