126 5.3 Limit Superior and Limit Inferior of Functions Remark 5.3.1 The infimum of the extended real-valued functiongon(0,∞) is defined as follows. If g(δ) =∞for all δ >0, then infδ>0g(δ) =∞. In the case where there exists δ0 >0 such that g(δ0) <∞, we have g(δ) ∈Rfor all 0<δ <δ0 and inf δ>0 g(δ) = inf 0<δ<δ0 g(δ) due to the increasing nature of gon (0,∞). We say that the function f is locally bounded around x0 if there exists δ >0 andM>0 such that | f (x)| ≤Mfor 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 everyε >0, there exists δ >0 such that f (x) < ℓ+ε for all x ∈B0(x0;δ)∩D; (ii) For everyε >0 and for every δ >0, there exists ˆ x ∈B0(x0;δ)∩Dsuch that ℓ−ε <f (ˆ x). Proof: Suppose ℓ =limsupx →x0 f (x). Then ℓ =inf δ>0 g(δ), where gis defined in (5.2). For anyε >0, there exists δ >0 such that ℓ ≤g(δ) = sup x∈B0(x0;δ)∩D f (x) < ℓ+ε. 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 x∈B0(x0;δ)∩D f (x). Thus, there exists ˆ x ∈B0(x0;δ)∩Dwith ℓ−ε <f (ˆ x). This proves (ii).
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