MthT 430 Chap 8h limsup and liminf for Functions

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(See also Spivak Chapter 8 - Problem 18) Let {x_k} be a bounded sequence. We define the limit superior and limit inferior of the sequence to be

limsup k → ∞  xk = lim k→∞  ⎛ ⎝ sup n ≥ k  xn ⎞ ⎠ , liminf k → ∞  xk = lim k→∞  ⎛ ⎝ inf n ≥ k  xn ⎞ ⎠ .

For the time being, we speak of a function f(x) defined and bounded near x=a. In the same spirit, we define limsup and liminf for functions.

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Let f be a bounded function. We define the limit superior and limit inferior of f near a to be

limsup x→ a  f(x) = lim δ→ 0+  ⎛ ⎝ sup 0 < |x−a| < δ  f(x) ⎞ ⎠ , liminf x→ a  f(x) = lim δ→ 0+  ⎛ ⎝ inf 0 < |x−a| < δ  f(x) ⎞ ⎠ .

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(P13-LUB) shows that both limsup_{x→ a} f(x) and liminf_{x→ a} f(x) exist.

Definition. If f is a bounded function defined near a, we define

M(f,a,δ) = sup 0 < |x−a| < δ  f(x), m(f,a,δ) = inf 0 < |x−a| < δ  f(x).

For δ > 0 and small enough, M(f,a,δ) is a bounded nondecreasing function of δ, and

lim δ→0+  M(f,a,δ) = inf δ > 0  M(f,a,δ)

exists.

For δ > 0 and small enough, m(f,a,δ) is a bounded nonincreasing function of δ, and

lim δ→0+  m(f,a,δ) = sup δ > 0  m(f,a,δ)

exists.

Necessary and Sufficient Condition (NASC) for Existence of a Limit of a Function

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Show that

limsup x→ a  f(x) = A

if and only if for every ϵ > 0,

⎧ ⎪ ⎨ ⎪ ⎩ there is a δ > 0 such that f(x) < A + ϵ for 0 < |x−a| < δ, for every δ > 0, there is an xδ, 0 < |xδ−a| < δ, such that f(xδ) > A − ϵ.

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Show that

liminf x→ a  f(x) = A

if and only if for every ϵ > 0,

⎧ ⎪ ⎨ ⎪ ⎩ there is a δ > 0 such that f(x) > A − ϵ for 0 < |x−a| < δ, for every δ > 0, there is an xδ, 0 < |xδ−a| < δ, such that f(xδ) < A + ϵ.

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Prove:

Theorem. Let f be a bounded function for x near a and ≠ a. Then

lim x→ a  f(x) exists

if and only if

liminf x→ a  f(x) = limsup x→ a  f(x)

Proof. Easy - If lim_{x→ a}f(x) = L, then liminf_{x→ a}f(x) = L, …. For the converse, use the first of the two conditions in the characterizations of limsup_{x→ a}f(x) = L and liminf_{x→ a}f(x) = L.

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On 18 Sep 2014, 21:28.