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

#22 Friday (11/01/96)

SUMMARY Continuity and Uniform Continuity

READING pages 87-93 §17, §19

I. Write the ϵ-δ definition of continuity of f(x) at x_{0}.

II. What quantities does your choice of δ depend on?

III. Let’s work specifically with f(x) = 1/x^{2}.

A. GEOMETRIC OBSERVATIONS

1. Where is this function continuous?

2. What are f(0.1), f(0.2) and f(0.25)? Mark these values on the y-axis of the accompanying graph.

3. From each of these values (f(0.1), f(0.2) and f(0.25)) on the y-axis, draw a horizontal line to the points (x, f(x)) on the graph, then vertical lines down to the x values (0.1, 0.2, 0.25). (I have done the lines for f(0.1) for you.)

4. Rewrite the definition of continuity using the function f(x) = 1/x^{2} and the point x_{0} = 0.2.

5. [Use a different color pen from 3.] Let ϵ = 5. Draw this ϵ-neighborhood around f(0.2) on the y-axis of the graph (i.e. draw a little interval from 5 units above f(0.2) to 5 units below). Now follow these y-values with horizontal lines over to the graph of 1/x^{2} and then down to points near x = 0.2 on the x-axis to see what corresponding δ-neighborhood you get. Using the picture, approximate what δ you must choose.

6. Find the exact δ we need to choose for ϵ = 5 by using a calculator and the function 1/x^{2}.

7. Repeat 5 & 6 with x_{0} = 0.1.

8. Repeat 5 & 6 with x_{0} = 0.25.

9. After this, do you change and/or believe your answer to II? (Explain the difference between your answers to 5/6, 7, and 8.)

B. ANALYTIC PROOF

1. Using the ϵ-δ definition of continuity, prove that f(x) = 1/x^{2} is continuous on the interval (0, ∞). Basically what I want you to do is figure out what your choice of δ should be for an arbitrary ϵ > 0. You’ll see the dependence on both x_{0} and ϵ! (Hint: When you get a “formula” for f(x) − f(x_{0}), assume that |x − x_{0}| < \frac{x_{0}}{2} and show that this implies |x + x_{0}| < \frac{5x_{0}}{2} .) Your work will be similar to Example 1(b) in the text.

2. FORMAL PROOF

Compare the choice of δ you get from this proof (say for ϵ = 5, x_{0} = 0.2) with what you got in 5 on the previous page.

IV. Let us say that f is UNIFORMLY CONTINUOUS on a set S when the choice of your δ does not depend on the points in S (the x_{0}). Try to write a definition for UNIFORM CONTINUITY on S.

Compare with Definition 19.1 in Ross. If it is different than your definition, then understand why...and make the appropriate changes above.

Why does it not make sense to talk about uniform continuity at a point?

Why must you talk about uniform continuity on a set?

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