126 5.3 Limit Superior and Limit Inferior of Functions Remark 5.3.1 The infmum of the extended real-valued function g on (0,∞) is defned as follows. If g(δ )= ∞ for all δ > 0, then infδ >0 g(δ )= ∞. In the case where there exists δ0 > 0 such that g(δ0) < ∞, we have g(δ ) ∈ R for all 0 < δ < δ0 and inf g(δ )= inf g(δ ) δ >0 0<δ <δ0 due to the increasing nature of g on (0,∞). We say that the function f is locally bounded around x0 if there exists δ > 0 and M > 0 such that | f (x)|≤ M for all x ∈ B0(x0;δ ) ∩ D. Clearly, if f is locally bounded around x0, then limsupx →x0 f (x) is a real number. Similar discussion applies for the limit inferior. Theorem 5.3.1 Let f : D → R, let x0 be a limit point of D, and let ℓ be a real number. Then ℓ = limsupx →x0 f (x) if and only if the following two conditions hold: (i) For every ε > 0, there exists δ > 0 such that f (x) <ℓ + ε for all x ∈ B0(x0;δ ) ∩ D; (ii) For every ε > 0 and for every δ > 0, there exists xˆ ∈ B0(x0;δ ) ∩ D such that ℓ − ε < f (xˆ). Proof: Suppose ℓ = limsupx →x0 f (x). Then ℓ = inf g(δ ), δ >0 where g is defned in (5.2). For any ε > 0, there exists δ > 0 such that ℓ ≤ g(δ )= sup f (x) <ℓ + ε. x∈B0(x0;δ )∩D Thus, f (x) <ℓ + ε for all x ∈ B0(x0;δ ) ∩ D, which proves condition (i). Next note that for any ε > 0 and δ > 0, we have ℓ − ε <ℓ ≤ g(δ )= sup f (x). x∈B0(x0;δ )∩D Thus, there exists xˆ ∈ B0(x0;δ ) ∩ D with ℓ − ε < f (xˆ). This proves (ii).
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