Introduction to Mathematical Analysis I - Second Edition

71 Example 3.2.6 It follows from Example 3.2.4 that lim x → 0 | x | x does not exists, since the one-sided limits do not agree. Definition 3.2.4 (monotonicity) Let f : ( a , b ) → R . (1) We say that f is increasing on ( a , b ) if, for all x 1 , x 2 ∈ ( a , b ) , x 1 < x 2 implies f ( x 1 ) ≤ f ( x 2 ) . (2) We say that f is decreasing on ( a , b ) if, for all x 1 , x 2 ∈ ( a , b ) , x 1 < x 2 implies f ( x 1 ) ≥ f ( x 2 ) . If f is increasing or decreasing on ( a , b ) , we say that f is monotone on this interval. Strict mono- tonicity can be defined similarly using strict inequalities: f ( x 1 ) < f ( x 2 ) in (1) and f ( x 1 ) > f ( x 2 ) in (2). Theorem 3.2.4 Suppose f : ( a , b ) → R is increasing on ( a , b ) and ¯ x ∈ ( a , b ) . Then lim x → ¯ x − f ( x ) and lim x → ¯ x + f ( x ) exist. Moreover, sup a < x < ¯ x f ( x ) = lim x → ¯ x − f ( x ) ≤ f ( ¯ x ) ≤ lim x → ¯ x + f ( x ) = inf ¯ x < x < b f ( x ) . Proof: Since f ( x ) ≤ f ( ¯ x ) for all x ∈ ( a , ¯ x ) , the set { f ( x ) : x ∈ ( a , ¯ x ) } is nonempty and bounded above. Thus, ` = sup a < x < ¯ x f ( x ) is a real number. We will show that lim x → ¯ x − f ( x ) = ` . For any ε > 0, by the definition of the least upper bound, there exists a < x 1 < ¯ x such that ` − ε < f ( x 1 ) . Let δ = ¯ x − x 1 > 0. Using the increasing monotonicity, we get ` − ε < f ( x 1 ) ≤ f ( x ) ≤ ` < ` + ε for all x ∈ ( x 1 , ¯ x ) = B − ( ¯ x ; δ ) . Therefore, lim x → ¯ x − f ( x ) = ` . The rest of the proof of the theorem is similar. Let B 0 ( ¯ x ; δ ) = B − ( ¯ x ; δ ) ∪ B + ( ¯ x ; δ ) = ( ¯ x − δ , ¯ x + δ ) \ { ¯ x } . Definition 3.2.5 (infinite limits) Let f : D → R and let ¯ x be a limit point of D . We write lim x → ¯ x f ( x ) = ∞ if for every M ∈ R , there exists δ > 0 such that f ( x ) > M for all x ∈ B 0 ( ¯ x ; δ ) ∩ D . Similarly, we write lim x → ¯ x f ( x ) = − ∞ if for every M ∈ R , there exists δ > 0 such that f ( x ) < M for all x ∈ B 0 ( ¯ x ; δ ) ∩ D .