125 (ii) bn −an = (b−a)/2 n−1, and (iii) f (an) <γ <f (bn). In this case, we proceed as follows. Condition (ii) implies that limn→∞(bn −an) =0. By the Nested Intervals Theorem (Theorem 2.3.2, part (b)), there exists c∈[a,b] such that T∞ n=1In ={c}. Moreover, as we see from the proof of that theorem, an →c andbn →c as n→∞. By the continuity of f , we get lim n→∞ f (an) =f (c) and lim n→∞ f (bn) =f (c). Since f (an) <γ <f (bn) for all n, condition (iii) above and Theorem 2.1.2 give f (c) ≤γ and f (c) ≥γ. It follows that f (c) =γ. Note that, since f (a) <γ < f (b), then c ∈(a,b). The proof is now complete. □ Exercises 5.2.1 ▷Let I be an interval and f : I →Rbe a continuous functions. Prove that f (I) is an interval. 5.3 Limit Superior and Limit Inferior of Functions We extend to functions the concepts of limit superior and limit inferior. For this it will be convenient to introduce a new notation for a ball without its center. Definition 5.3.1 Given x0 ∈Randδ >0, we denote byB0(x0,δ) the set B0(x0;δ) =B(x0;δ)\ {x0}=B−(x0;δ)∪B+(x0;δ) = (x0 −δ,x0)∪(x0,x0 +δ). With this notation the point x0 is a limit point of the set Dif for all δ >0, the set B0(x0;δ)∩D̸=0/ . Definition 5.3.2 Let f : D→Rand let x0 be a limit point of D. The limit superior of the function f at x0 is defined by limsup x→x0 f (x) =inf δ>0 sup x∈B0(x0;δ)∩D f (x). Similarly, the limit inferior of the function f at x0 is defined by liminf x→x0 f (x) =sup δ>0 inf x∈B0(x0;δ)∩D f (x). Consider the extended real-valued function g: (0,∞) →R∪{∞}defined by g(δ) = sup x∈B0(x0;δ)∩D f (x). (5.2) It is clear that gis increasing and limsup x→x0 f (x) =inf δ>0 g(δ).
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