71 Defnition 3.2.4 (monotonicity) Let f : (a,b) → R. (i) We say that f is increasing on (a,b) if, for all x1,x2 ∈ (a,b), x1 < x2 implies f (x1) ≤ f (x2). (ii) We say that f is decreasing on (a,b) if, for all x1,x2 ∈ (a,b), x1 < x2 implies f (x1) ≥ f (x2). If f is increasing or decreasing on (a,b), we say that f is monotone on this interval. Strict monotonicity can be defned similarly using strict inequalities: f (x1) < f (x2) in (i) and f (x1) > f (x2) in (ii). Theorem 3.2.4 Suppose f : (a,b) → R is increasing on (a,b) and x0 ∈ (a,b). Then lim − f (x) x →x0 and lim + f (x) exist. Moreover, x →x0 sup f (x)= lim f (x) ≤ f (x0) ≤ lim f (x)= inf f (x). x→x− x→x+ x0<x<b a<x<x 0 0 0 Proof: Since f (x) ≤ f (x0) for all x ∈ (a,x0), the set { f (x) : x ∈ (a,x0)} is nonempty and bounded above. By completeness axiom, the supremum of the set exists, say ℓ = sup{ f (x) : x ∈ (a,x0)}. We will show that limx →x− 0 f (x)= ℓ. For any ε > 0, by the defnition of the least upper bound, there exists a < x1 < x0 such that ℓ − ε < f (x1). Let δ = x0 − x1 > 0. Using the increasing monotonicity, we get ℓ − ε < f (x1) ≤ f (x) ≤ ℓ<ℓ + ε for all x ∈ (x1,x0)=(x0 − δ ,x0). Therefore, limx →x− 0 f (x)= ℓ. The rest of the proof of the theorem is similar. □ Defnition 3.2.5 (infnite limits) Let f : D → R and let x0 be a limit point of D. (i) We say that f has limit ∞ as x → x0 if for every M ∈ R, there exists δ > 0 such f (x) > M for all x ∈ D for which 0 < |x − x0| < δ and write limx →x0 f (x)= ∞, (ii) We say that f has limit −∞ as x → x0 if for every L ∈ R, there exists δ > 0 such f (x) < L for all x ∈ D for which 0 < |x − x0| < δ and write limx →x0 f (x)= −∞, Infnite limits of functions have similar properties to those of sequences from Chapter 2 (see Defnition 2.3.2 and Theorem 2.3.4). ■ Example 3.2.7 We prove from the defnition that 1 lim = ∞. x→0 x2 Let M ∈ R be given. We can assume M > 0 because if the inequality f (x) > M is true for a positive M then it is also true for all numbers less or equal to M. We want to fnd δ > 0 that will 1 guarantee f (x)= x2 > M whenever 0 < |x| < δ . As in the case of fnite limits, we work backwards from f (x) > M to fnd a suitable δ .
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