Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

#25 Monday (11/17/96)

SUMMARY Uniform Continuity

READING §19 (page 102-110)

\underline{Warm-Up} We’ll review Exam 2.

Let’s show that f(x) = 1/x^{2} is not uniformly continuous on (0, ∞) even though we know that it is uniformly continuous on (a, ∞), a > 0.

READ EXAMPLE 3 on page 102 of Ross

Uniform Continuity 19.2 If f is continuous on a closed interval [a,b], then f is uniformly continuous on [a, b].

USEFULNESS OF UNIFORM CONTINUITY

\underline{Theorem} 19.4 If f is uniformly continuous on a set S and (s_{n}) is a Cauchy sequence in S, then (f(s_{n})) is a Cauchy sequence.

This theorem can be used to show (again) that f(x) = 1/x^{2} is not uniformly continuous on (0, 1)

\underline{Definition} A function f̃ is said to be an extension of f if dom(f) ⊆ dom(f̃) and f(x) = f̃(x) for all x ∈ dom(f) If f(x) = sin(1/x) for x ∈ (0, 1/π] Observe this extension of f written below:

f̃(x) = \{ \begin{matrix}x sin(1/x) & for & 0 < x ≤ 1/π \\ 0 & for & x = 0 \end{matrix}

What is dom(f)? What is dom(f̃)? Is f̃ continuous at x = 0?

\underline{Theorem} 19.5 A real-valued function on (a,b) is uniformly continuous on (a,b) if and only if it can be extended to a continuous function f̃ on [a,b]

Homework Assignment (Due Friday November 22) §19, page 109: 19.1, 19.2, 19.4(a), 19.5(d),(e),(f)