Comments on the textbook

Exercise 1.3.3

The discussion is incomplete. One needs to check additionally that the supposed subfield contains the additive and multiplicative identity elements and contains for each of its nonzero elements the additive and multiplicative inverses. The same comment applies to Note 1 at the end of the chapter.

Section 1.10

In the paragraph titled “Properties of the Distance Function”, item 3, there is a duplicated closing right-hand parenthesis at the end of the sentence.

Note 7 for Chapter 1

For (x2y2)  read (x2y2) 

Exercise 2.9.1

In part (a), the right-hand side should be sn+1− 

Section 2.11

In the paragraph preceding Theorem 2.40, one of the date ranges is set with a hyphen and the other with an en dash. The latter punctuation is standard.

Section 2.12

In Example 2.42, sixth sentence, for “the N stage” read either “stage N” or “the Nth stage”.

Chapter 2, Note 26

For 16 read 116

Chapter 2, Note 29

In the second displayed formula, the expression y−n+2m is intended to be y−(n+2m) (although the sign change does not much matter, since m is arbitrary).

Section 4.1

In the second sentence, Heine's personal name should be spelled “Eduard” according to the discussion just above Definition 4.28 later in the chapter.

Section 4.2.3

In Example 4.6, item 1, there is a spurious doubled right-hand parenthesis at the end of the sentence.

Section 4.3.1

In Example 4.10, item 2, insert the word “is” preceding “not closed”.

Exercise 4.3.23

In part (b), add a question mark at the end of the first sentence.

Section 4.5, proof of Theorem 4.33

In the second paragraph, second sentence, the (correct) statement “there exists (x)0 for which (x−tx+t)Ux for all t(0(x))” is a convoluted way of saying “there exists a positive number (x) such that the interval (x−(x)x+(x)) is a subset of Ux”. The intended statement is probably the equivalent statement “there exists (x)0 for which [x−tx+t]Ux for all t(0(x))” (for the construction of a Cousin cover requires the consideration of closed intervals). A similar comment applies to the sixth sentence in the paragraph.

Exercise 4.5.18

The name of the Finnish mathematician Lindelöf is misspelled with a doubled terminal consonant. The same error appears in Exercise 4.5.19 and in Note 83 at the end of the chapter.

Exercise 4.7.4

The four statements do not parse. Part (a) should begin, “Let A=[01] Describe […]”, and similarly for the other parts.

Exercise 4.7.10

This exercise is identical to Exercise 4.5.10.

Exercise 4.7.15

In part (d), the word “interval” has to be interpreted to allow, as a special case, a single point (a degenerate interval).

Section 5.2.3

In Exercise 5.2.12, insert a period between 2 and “Define”. And after Exercise 5.2.16, delete the orphaned period.

Section 5.2.4

In the first line of the proof of Corollary 5.24, for “first of the these” read “first of these”.

Section 5.2.5

At the end of the first paragraph, the claimed falsity arises only because one or both sides of the equation can be undefined. See also Exercise 5.2.23.

Section 5.2.6

Most authors use the terminology “Dirichlet function” to refer to the characteristic function of the rational numbers, since Dirichlet considered the example of a function that takes a constant value c on the rationals and a different constant value d on the irrationals [Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données, Journal für die reine und angewandte Mathematik 4 (1829) 157–169; see p. 169]. The function called “Dirichlet function” in the text seems to be due to Karl Johannes Thomae [Einleitung in die Theorie der bestimmten Integrale, 1875, p. 14] and is commonly known as “the ruler function”.

Exercises 5.4.7 and 5.4.8

The punctuation mark at the end should be a period, not a comma.

Section 5.9.3

In the proof of Theorem 5.64, third paragraph, the pointsa and b such that abc are not the points a and b in the statement of the theorem.

Chapter 5, Notes

In Note 107, there is no need to assume positivity, for the exercise has absolute values in it.

In Note 109, for “sum and products rule” read “sum and product rules”.

Note 118 does not respond to the statement of the exercise, for there is no “same collection” at hand, and applying Cousin's lemma is not in the spirit of the instruction to “[a]djust the proof of Theorem 5.48”. Actually, the given proof of Theorem 5.48 carries over almost word for word.

In Note 137, the claim that “[t]he function must be onto” is false. For instance, the function might be a constant function. The statement of the exercise is nonetheless true. You can apply the intermediate-value property to the function that sends x to f(x)−x

Section 7.2.1

The parenthetical remark preceding Example 7.4 needs a terminal period.

Section 7.3.1

Equation (4) in the proof of Theorem 7.7 has an error: the term −f(x) should be −f(x0)

In Exercise 7.3.2, for “Figure fig-table2” read “Figure 7.3”.

Section 7.3.3

In the second paragraph, second sentence, for “that that” read “that”.

The argument in the third paragraph does not prove what is claimed. The calculation determines the derivative of the inverse function under the hypothesis that the inverse function is differentiable. But what is claimed is that the inverse function is, in fact, differentiable. That fact is proved later in Theorem 7.32 of Section 7.9.

Exercise 7.4.2

In part (a), both instances of f without a subscript should be fn

Exercise 7.6.2

The statement of the exercise is incorrect, for the equation

x3+3x2−4

has both −2 and 1 as solutions. The statement of the exercise can be corrected by replacing “cannot have more than one solution” with “cannot have more than one positive solution”. Alternatively, add the hypothesis that 0.

Exercise 7.6.8

The unidentified letter M denotes a fixed positive real number (a constant).

Exercise 7.6.18

The statement of the exercise is incorrect. If f(x)=x2, then f has a continuous second derivative, yet the indicated limit is equal to 2 not 0 The intended statement is that the limit exists, not that the limit equals 0

Section 7.7

In the Note following Definition 7.22, third sentence, delete the unmatched closing right-hand parenthesis.

Section 7.8

Example 7.29 is incorrectly stated. The function g is supposed to be the characteristic function of the rational numbers, not the irrationals.

In the proof of Theorem 7.30, first displayed formula, delete the spurious closing right-hand parenthesis preceding the closing right-hand bracket.

In the proof of Theorem 7.30, third paragraph, insert a space after the period that ends the first sentence.

Section 7.9

In the second paragraph, third sentence, “the extreme value” is misleading, for there is not a unique extremum. The point is that if f(a)0f(b), then the global minimum cannot occur at an endpoint and hence occurs at an interior point; if f(b)0f(a), then the global maximum cannot occur at an endpoint and hence occurs at an interior point.

Section 7.10

In the line preceding equation (14), insert a period before “Then”.

In Exercise 7.10.13, some hypothesis on the set A is needed to guarantee convergence of the series; boundedness ofA will do.

Section 7.11

In the paper version, the section titles and the page headers are missing the letter ô in the name L'Hôpital. The pdf version for screen viewing is correct.

Example 7.37, second paragraph, two-line display, the derivativeg(x) should be 3+15x2, not 3+5x2

Proof of Theorem 7.38, third paragraph, third sentence, “contradicting hypothesis (ii)” should say “contradicting hypothesis (iii)”.

Proof of Theorem 7.41, last line, add a period.

In Exercise 7.11.1(c), the terminal punctuation should be a question mark.

Exercise 7.13.1

The given condition contains no information about the value of the function f at the point x0 so the property cannot possibly be equivalent to differentiability of f at x0 The property is, however, equivalent to the statement that f has a removable singularity at x0 and the function obtained by removing the singularity is differentiable at x0 An analogous issue arises in Exercise 7.13.2.

Exercise 7.13.11

The name “Chisholm” is missing the letter “h”.

Chapter 7, Notes

In Note 187 for Exercise 7.6.8, the name Lipschitz is misspelled (the letter c is missing).

In Note 201 for Exercise 7.8.9, the words “For an example” should go with the first sentence, not the second sentence. (The first sentence of the hint pertains to the second part of the exercise, while the second sentence of the hint pertains to the first part of the exercise.)

In Note 202 for Exercise 7.8.10, some hypothesis is needed on the set A to guarantee convergence of the series; boundedness ofA will do.

In Note 203 for Exercise 7.9.1, the condition f(0)=0 is supposed to be f(0)=0

In Note 211 for Exercise 7.10.14, the punctuation at the end of the second line of the three-line display is awry.

In Note 215 for Exercise 7.11.10, the hint should say that

limxf(x)g(x)=0

The denominator g(x) is missing.

Section 8.2

The first sentence of Corollary 8.3 is missing its terminal period.

Exercise 8.3.1

In the first displayed formula, delete the spurious vertical bar following [ab]

Section 8.6

In Example 8.14, end of first paragraph, for “and to” read “and”.

In the proof of Theorem 8.17, fourth displayed equation, the closing right-hand parenthesis is missing after the argument of f.

Exercise 8.10.1

In the first displayed formula, infm(f) should be supm(f)

Chapter 8, Notes

In Note 239 for Exercise 8.9.2, delete the spurious letter “x” at the end of the line.

Appendix A

In Section A.2, the paragraph about Set-Builder Notation, delete the spurious closing right-hand parenthesis at the end of the second sentence.

In Section A.5, third paragraph, the Latin wordreductio is a noun, not a verb, so the translation is “reduction”, not “I reduce”. In the subsequent indented paragraph, there is a spurious space between the symbol Q and the period; in the pdf version, the period consequently has gotten displaced to a separate line.

At the end of Exercise A.8.8, the phrase “well ordering of ” is mysteriously repeated.

In Section A.9, end of second paragraph, the phrase “there do exist numbers” should be in the indicated example “there does exist a number”.

In Note 388, the closing delimiter after the date range should be a parenthesis, not a square bracket.

Harold P. Boas