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

#21 Wednesday (10/30/96)

SUMMARY Continuity

READING pages 87-93 §17

\underline{Warm-Up} (5-10 minutes) Draw a picture of a continuous function and a picture of a discontinuous function

Can you think of a way to link the idea of continuity with the idea of sequences? \underline{Definition} 17.1 Let f be real-valued function whose domain dom(f) ⊆ ℝ. The function f is said to be continuous at x_{0}, where x_{0} ∈ dom(f), if for EVERY sequence (x_{n}) in dom(f) where lim x_{n} = x_{0} we have lim f(x_{n}) = f(x_{0})

The function f is said to be continuous if it is continuous at EVERY point in dom(f).

\underline{Definition} 17.2 Let f be real-valued function whose domain is a subset of ℝ. Then f is continuous at x_{0}, where x_{0} ∈ dom(f), if and only if

∀ϵ > 0 ∃δ > 0 ∋ x ∈ dom(f) and |x − x_{0}| < δ ⇒ |f(x) − f(x_{0})|

\underline{Example} Show that f(x) = x^{2} + 1 is continuous at x = 2

Consider the function f(x) = x^{2} sin(1/x) for x ≠ 0 and f(0) = 0 Use Derive to graph this function. Does it LOOK continuous at x = 0?

\underline{Exercise} Show that f(x) is continuous at x = 0.

\underline{Exercise} Show that if f(x) = (1/x) sin(1/x^{2}) for x ≠ 0 and f(0) = 0 then this function is discontinuous at x = 0. Use Derive to graph this function. Does it LOOK continuous at x = 0?

Homework Assignment (Due Monday November 4) §17, page 56: 17.1, 17.3(a),(c),(g), 17.4, 17.9(b),(c), 17.10(b)