126 5.3 Limit Superior and Limit Inferior of Functions Remark 5.3.1 The infimum of the extended real-valued functiong on(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 g on(0,∞). We say that the function f is locally bounded around x0 if there exists δ >0andM>0 such that | f (x)|≤Mfor all x ∈B0(x0;δ)∩D. Clearly, if f is locally bounded around x0, then lim supx →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 every ε >0 and for everyδ >0, there exists ˆx ∈B0(x0;δ)∩Dsuch that ℓ−ε <f (ˆx). Proof: Suppose ℓ =limsupx →x0 f (x). Then ℓ =inf δ>0 g(δ), where g is 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 anyε >0 and δ >0, we have ℓ−ε <ℓ ≤g(δ)= sup x∈B0(x0;δ)∩D f (x). Thus, there exists ˆx ∈B0(x0;δ)∩Dwith ℓ−ε <f (ˆx). This proves (ii).
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