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 T∞ Intervals Theorem (Theorem 2.3.2, part (b)), there exists c ∈ [a,b] such that =1 In = {c}. Moreover, n as we see from the proof of that theorem, an → c and bn → c as n → ∞. By the continuity of f , we get lim f (an)= f (c) and n→∞ lim f (bn)= f (c). n→∞ 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 → R be 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. Defnition 5.3.1 Given x0 ∈ R and δ > 0, we denote by B0(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 D if for all δ > 0, the set B0(x0;δ )∩D ≠ 0/. Defnition 5.3.2 Let f : D → R and let x0 be a limit point of D. The limit superior of the function f at x0 is defned by limsup f (x)= inf sup f (x). x→x0 δ >0 x ∈B0(x0;δ )∩D Similarly, the limit inferior of the function f at x0 is defned by liminf f (x)= sup inf f (x). x→x0 x∈B0(x0;δ )∩D δ >0 Consider the extended real-valued function g: (0,∞) → R∪{∞} defned by g(δ )= sup f (x). (5.2) x∈B0(x0;δ )∩D It is clear that g is increasing and limsup f (x)= inf g(δ ). δ >0 x→x0
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