Introduction to Mathematical Analysis I - Second Edition

90 3.6 LIMIT SUPERIOR AND LIMIT INFERIOR OF FUNCTIONS Consider the extended real-valued function g : ( 0 , ∞ ) → ( − ∞ , ∞ ] defined by g ( δ ) = sup x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) . (3.10) It is clear that g is increasing and limsup x → ¯ x f ( x ) = inf δ > 0 g ( δ ) . We say that the function f is locally bounded above around ¯ x if there exists δ > 0 and M > 0 such that f ( x ) ≤ M for all x ∈ B ( ¯ x ; δ ) ∩ D . Clearly, if f is locally bounded above around ¯ x , then limsup x → ¯ x f ( x ) is a real number, while limsup x → ¯ x f ( x ) = ∞ in the other case. Similar discussion applies for the limit inferior. Theorem 3.6.1 Let f : D → R and let ¯ x be a limit point of D . Then ` = limsup x → ¯ x f ( x ) if and only if the following two conditions hold: (1) For every ε > 0, there exists δ > 0 such that f ( x ) < ` + ε for all x ∈ B 0 ( ¯ x ; δ ) ∩ D ; (2) For every ε > 0 and for every δ > 0, there exists x δ ∈ B 0 ( ¯ x ; δ ) ∩ D such that ` − ε < f ( x δ ) . Proof: Suppose ` = limsup x → ¯ x f ( x ) . Then ` = inf δ > 0 g ( δ ) , where g is defined in ( 3.10 ) . For any ε > 0, there exists δ > 0 such that ` ≤ g ( δ ) = sup x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) < ` + ε . Thus, f ( x ) < ` + ε for all x ∈ B 0 ( ¯ x ; δ ) ∩ D , which proves condition (1) . Next note that for any ε > 0 and δ > 0, we have ` − ε < ` ≤ g ( δ ) = sup x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) . Thus, there exists x δ ∈ B 0 ( ¯ x ; δ ) ∩ D with ` − ε < f ( x δ ) . This proves (2) .