Comments on Chapter 3

Page 69

Notice that by taking \(\delta\) equal to \(\epsilon/(|m|+1)\), one could avoid introducing a special case.

Page 73, Theorem 3.10

The hypothesis “\(f\) and \(g\) have a limit” would be clearer if it said “both \(f\) and \(g\) have a limit” or “\(f\) and \(g\) have limits”, for there are potentially two different limits involved.

Page 74, Exercise 3.1.1

In standard usage, the pronoun “each” takes a singular verb, so the instruction should be either “prove that each of the following limits exists” or “prove that the following limits exist”.

Although there is only one meaningful interpretation of the formula in part a, it would be clearer to insert parentheses to make it read \[\lim_{x\to 2} (x^2 +2x -5)=3.\] A similar comment applies to part c.

Page 75, Exercise 3.1.3e

By convention, the problem is to be interpreted as \[\lim_{x\to 0} (\tan x) \sin\left( \frac{1}{x^2} \right) \] and not as \[\lim_{x\to 0} \tan\left( x \sin\left(\frac{1}{x^2} \right)\right). \] (It happens in this instance that the two different expressions have the same limit.)

Page 77, Definition 3.12

Other notations for the right-hand limit are \(f(a+0)\) and \(f(a^{+})\) (with a superscript).

Page 78

In the last line of the proof of Theorem 3.14, the cited formula (1) is the definition of limit from back on page 68.

Page 78, Definition 3.15

There is a typographical error in the second set of displayed formulas in part i: both instances of \(\infty\) should be \(-\infty\).

Page 79

Notice that the second paragraph from the bottom of the page implicitly generalizes the definition of the word “endpoint”. Originally (on page 13) this word applied only to intervals that are bounded from both sides. Under the new, extended meaning of the word, both \(a\) and \(\infty\) are endpoints of both of the intervals \((a,\infty)\) and \([a,\infty)\).

Page 80

The first line of the proof of Theorem 3.17 should say “already proved this for finite two-sided limits”.

Page 82

In the special case of Exercise 3.2.3a in which both \(a\) and \(n\) are equal to 0, you have to interpret the indeterminate expression \(0^0\) as being \(1\).

In Exercise 3.2.5, the hypothesis “\(a\in\mathbf{R}\)” is redundantly repeated.

Page 83, proof of Remark 3.20

It is not strictly true that “condition (5) is identical to (1)” (where the latter condition is back on page 68), for condition (1) has the restriction that \(0\lt |x-a| \lt \delta\), while (5)  has the restriction that \(|x-a|\lt \delta\). The difference between these two conditions is important to remember in general. In the present context, however, the two conditions are only superficially different: if \(|x-a|=0\), then \(|f(x)-f(a)|=0\), and \(0\) is certainly less than an arbitrary positive \(\epsilon\).

Page 84, Definition 3.23

The hypothesis “\(f(A)\subseteq B\) for every \(x\in A\)” should say simply “\(f(A)\subseteq B\)”, or alternatively “\(f(x)\in B\) for every \(x\in A\)”.

Page 88, proof of Example 3.31

There is nothing wrong with the proof, but notice that the proof of Example 3.7 back on page 71 already shows that \(f(0{+})\) fails to exist. Since \(\sin(1/x)\) is an odd (antisymmetric) function, it follows immediately that \(f(0{-})\) fails to exist.

Page 88, proof of Example 3.32

The unstated punchline of the argument is that the definition of continuity at \(a\) is contradicted when \(\epsilon\) equals \(1/2\) (say).

Page 89, Example 3.33

The function in this example is commonly known as “the ruler function”.

Page 91, Exercise 3.3.6

The adverb “nowhere” potentially could modify either the verb “exist” or the adjective “continuous”, so a hyphen is needed to prevent misreading: “there exist nowhere-continuous functions”.

Harold P. Boas