Assignment 4

Reading and Notes Questions

Question 1.  State the official definition of the sentence “f has a limit L at x_0” that appears in our textbook.

Question 2.  Rephrase the definition from Question 1. using the language and notation of “\varepsilon-balls” and “punctured \delta-balls.”

Question 3.  If we write \displaystyle \lim_{x\to x_0} \, f(x) = L, then what (if anything) can we conclude about f(x_0)?

Question 4.  On pages 65-66, our textbook explores the function f(x) = x/|x|.  Does their discussion prove that the limit of f(x) at x = 0 exists or doesn’t exist?  What style of proof did they use in this discussion?

Question 5.  In the textbook’s discussion of Example 2.5 (pages 68-69), the line “However, for a fixed positive integer q, there are but a finite number of points in [0, 1] of the form p/q.  In fact, for a fixed positive number q_0, there are only a finite number of points in [0, 1] of the form p/q where q \leq q_0 with p and q positive integers.”

Explain why this line is true.  Also, the function being discussed has a name, it is _______’s function.  Find the name!

Question 6.  Draw a visual interpretation of Theorem 2.1 (on page 69).  Why in the statement of this theorem is it stipulated that x_n \neq x_0?  In the discussion before the proof of this theorem, what previous homework problem was used?

Question 7.  Write a very short (but rigorous!) proof of Theorem 2.2 (you may, of course, assume Theorem 2.1 is true).

Question 8.  What is a less formal way to express the content of Theorem 2.3?

Question 9.  Write a poem about Example 2.6.

Question 10.  Explain exactly where Theorem 2.4 was used in Example 2.7.

Question 11.  In the definition of “limit of a function” (on page 64), is our textbook actually claiming that x_0 is an interior point for the domain D of f?  Explain your answer.

Proof Questions

Problem 1.  Prove or disprove:  the set \mathbb{Q} \subset \mathbb{R} is open.

Problem 2.  (a) Complete the following proof :

Suppose f:D \to \mathbb{R} is a function with x_0 an accumulation point of D, and assume L_1 and L_2 are limits of f at x_0 (in accordance with the definition on page 64).

Let \varepsilon > 0 be given.  We aim to show that |L_1 - L_2| < \varepsilon, which will complete our proof.

By definition of limit, there exists a ??? > 0 so that

\displaystyle 0 < \left|x - x_0\right| < \text{???} \Rightarrow \left|f(x) - L_1\right| < \frac{\varepsilon}{\text{??}}

\displaystyle 0 < \left|x-x_0\right| <\text{???} \Rightarrow \left|f(x) - L_2\right| < \frac{\varepsilon}{\text{??}}

(assuming x \in D, too).  Let such a ???? > 0 be chosen, and let x \in D and x \in B_{\text{??}}(x_0).  Then

\displaystyle \left|L_1 - L_2\right| = \left|L_1 - f(x) + f(x) - L_2\right|

\displaystyle \leq \left|L_1 - f(x)\right| + \left|L_2 - f(x)\right| < \frac{\varepsilon}{\text{??}} + \frac{\varepsilon}{\text{??}} = \varepsilon

as desired.  Therefore, L_1 _____  L_2.    \square

(b) This proof should feel suspiciously similar to the proof for a previous homework problem.  Which one was/is it?

(c ) Is it possible for a function f:D\to\mathbb{R} have two different limits at a single point x_0?

Problem 3.  Complete exercise 11. on page 79.

Problem 4.  Complete exercise 16. on page 80 (citing relevant theorems from Section 2.3).

Problem 5.  Prove that any function of the form f(x) = mx + b where m, b \in \mathbb{R} are constants is continuous at every point in \mathbb{R}.

Problem 6.  Prove that the set S = (-10, 0] is neither open nor closed.

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