71 Definition 3.2.4 (monotonicity) Let f : (a,b) →R. (i) We say that f is increasingon (a,b) if, for all x1,x2 ∈(a,b), x1 <x2 implies f (x1) ≤f (x2). (ii) We say that f is decreasingon (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 defined 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) →Ris increasing on (a,b) and x0 ∈(a,b). Then limx →x− 0 f (x) and limx →x+ 0 f (x) exist. Moreover, sup a<x<x0 f (x) = lim x→x− 0 f (x) ≤f (x0) ≤ lim x→x+ 0 f (x) = inf x0<x<b f (x). 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, sayℓ =sup{f (x) : x ∈(a,x0)}. We will show that limx →x− 0 f (x) =ℓ. For anyε >0, by the definition 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. □ Definition 3.2.5 (infinite limits) Let f : D→Rand 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) >Mfor all x ∈Dfor 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) <Lfor all x ∈Dfor which 0<|x−x0| <δ and write limx→x0 f (x) =−∞, Infinite limits of functions have similar properties to those of sequences from Chapter 2 (see Definition 2.3.2 and Theorem 2.3.4). ■ Example 3.2.7 We prove from the definition that lim x→0 1 x2 =∞. Let M∈Rbe given. We can assume M>0 because if the inequality f (x) >Mis true for a positive Mthen it is also true for all numbers less or equal to M. We want to find δ >0 that will guarantee f (x) = 1 x2 >Mwhenever 0<|x| <δ. As in the case of finite limits, we work backwards from f (x) >Mto find a suitable δ.
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