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

#24 Wednesday (11/13/96)

SUMMARY Uniform Continuity and Exam 2 Review

READING §19 (page 100-102)

\underline{Warm-Up} Recall the proof of how to show that f(x) = 1/x^{2} is continuous at x = x_{0}, where x_{0} ∈ (0, ∞)

Is there a way to pick δ so that it does NOT depend on x_{0}? Then we would say that f(x) is UNIFORMLY CONTINUOUS on (0, ∞) Uniform Continuity 19.1 Let f be a real-valued function whose domain is a subset S of ℝ . Then f is uniformly continuous on S, if and only if

∀ϵ > 0 ∃δ > 0 ∋ x, y ∈ S and |x − y| < δ ⇒ |f(x) − f(y)| < ϵ

\underline{Exercise} Show that if f(x) = 1/x^{2} is uniformly continuous on (a, ∞) where a is some number a > 0

\underline{Example} Show that f(x) = 3x^{2} + 1 is uniformly continuous on [−2, 2].