Introduction to Mathematical Analysis I - Second Edition

89 (a) f ( x ) = x sin ( 1 x ) on ( 0 , 1 ) . (b) f ( x ) = x x + 1 on [ 0 , ∞ ) . (c) f ( x ) = 1 | x − 1 | on ( 0 , 1 ) . (d) f ( x ) = 1 | x − 2 | on ( 0 , 1 ) . 3.5.4 Let D ⊂ R and k ∈ R . Prove that if f , g : D → R are uniformly continuous on D , then f + g and k f are uniformly continuous on D . 3.5.5 Give an example of a subset D of R and uniformly continuous functions f , g : D → R such that f g is not uniformly conitnuous on D . 3.5.6 Let D be a nonempty subset of R and let f : D → R . Suppose that f is uniformly continuous on D . Prove that if { x n } is a Cauchy sequence with x n ∈ D for every n ∈ N , then { f ( x n ) } is also a Cauchy sequence. 3.5.7 B Let a , b ∈ R and let f : ( a , b ) → R . (a) Prove that if f is uniformly continuous, then f is bounded. (b) Prove that if f is continuous, bounded, and monotone, then it is uniformly continuous. 3.5.8 B Let f be a continuous function on [ a , ∞ ) . Suppose lim x → ∞ f ( x ) = c . (a) Prove that f is bounded on [ a , ∞ ) . (b) Prove that f is uniformly continuous on [ a , ∞ ) . (c) Suppose further that c > f ( a ) . Prove that there exists x 0 ∈ [ a , ∞ ) such that f ( x 0 ) = inf { f ( x ) : x ∈ [ a , ∞ ) } . 3.6 LIMIT SUPERIOR AND LIMIT INFERIOR OF FUNCTIONS We extend to functions the concepts of limit superior and limit inferior. Definition 3.6.1 Let f : E → R and let ¯ x be a limit point of D . Recall that B 0 ( ¯ x ; δ ) = B − ( ¯ x ; δ ) ∪ B + ( ¯ x ; δ ) = ( ¯ x − δ , ¯ x ) ∪ ( ¯ x , ¯ x + δ ) . The limit superior of the function f at ¯ x is defined by limsup x → ¯ x f ( x ) = inf δ > 0 sup x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) . Similarly, the limit inferior of the function f at ¯ x is defined by liminf x → ¯ x f ( x ) = sup δ > 0 inf x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) .