Comments on Chapter 5

Page 130

The expression \( \{x_0\), …, \(x_n\}\) for a partition should have a comma following the ellipsis dots. The comma is missing in lines 2, 5, and 7 of Definition 5.1, line 1 of Definition 5.3 on page 131, line 1 of the proof of Remark 5.7 on page 132, and the line above equation (17) on page 157.

Page 139

Exercise 5.1.4a has a hint in the back of the book that refers to the “Sign Preserving Property”. I don't know where this property is stated in the book, but you could look either at Lemma 3.28 on page 86 or at the Comparison Theorem that is coming up on page 145.

Page 141

In Definition 5.17i, the notation in the displayed formula may be confusing, since the index \(j\) is free on the left-hand side and bound on the right-hand side. It would perhaps be clearer to indicate in the notation that there is a set of samples by writing \(S(f,P,\{t_j\})\).

Page 142

At lines 1–3, the Approximation Property furnishes two partitions, one for the lower sum and one for the upper sum. You can take \(P_\epsilon\) to be a common refinement of these two partitions. The same device applies at the beginning of the proof of Theorem 5.19 on the next page.

Page 143

At the end of the first paragraph of the proof of Theorem 5.19, the unstated punchline is that you can invoke the linearity property of finite sums, a property that you know very well (and can easily prove by induction).

Page 145

At line 4, delete the reference to Exercise 5.2.4 (which does not apply).

Page 148

In the proof of Theorem 5.26, one should, of course, shrink \(\delta\) to ensure that \(x_0+\delta\lt b\).

In lines 4 and 6 of Theorem 5.27, for “an \(c\)” read “a \(c\)”. The same comment applies to lines 3 and 10 on the next page.

Page 149

In the last sentence, “\(M\) is the maximum value” should say “\(M\) is any number greater than or equal to the maximum value” (which is the situation illustrated in Figure 5.4).

Page 150

Exercise 5.2.2c is wrong. For example, suppose \(n=1\) and \(f(x)=x+2\). If \(a=-1\) and \(b=-1+\sqrt{3}\), then \(f(x)\) is never equal to \(0\) on the interval \([a,b]\), yet \(\int_{a}^{b} f(x)x\,dx=0\).

Page 156

In formula (15), \(\Delta t_j\) should be \(\Delta x_j\). In the last displayed equation on the page, \(\Delta x_j\) should be \(\Delta t_j\), while \( \phi(t_j)-\phi(t_{j-1})\) should be \(\phi(x_j)-\phi(x_{j-1})\).

Page 157

A preposition is missing in line 2: make it “Riemann sum of the left side”. In the displayed equation in the next line, \(\widetilde{P}\) should be just \(P\).

Page 158

In the sentence following the four-line display, the two instances of \(P\) should be \(\widetilde{P}\). Observe that there is a natural bijection between refinements of the partition \(P\) and refinements of the partition \(\widetilde{P}\).

In line 2 of Case 2, \(\phi^{-1}(x_n)\) and \(\phi^{-1}(x_0)\) should be \(\phi^{-1}(t_n)\) and \(\phi^{-1}(t_0)\).

Page 161

Exercise 5.3.2a is correct, but a stronger statement holds: namely, \(\int_1^4 f(\sqrt{x})\,dx=12\).

Page 162

The Warning on page 32 says that the meaning of \(L^{-1}\) is to be determined from context. In Exercise 5.3.8, the context is indicated by part a, from which you should infer that \(L^{-1}\) here means the inverse function (not the reciprocal). The function \(E\) is, of course, the standard exponential function.

Page 163, Exercise 5.3.11

The exercise is wrong, even when \(q=1\), since modifying the value of the function \(f\) at one point leaves the left-hand side of the equation unchanged.

Page 164, Definition 5.38

When \(b=\infty\), the notation \(\lim_{d\to b{-}}\) has not officially been defined, but of course the meaning then is \(\lim_{d\to \infty}\). Equation (18) could be rephrased using the notation introduced in equation (4) on page 79.

Page 166

In the first clause of line 3, we can say only that \(F\) is increasing on \([c,b)\), not on \([c,b]\), since we do not yet know that the function \(F\) is defined at the point \(b\).

The proof of Remark 5.46 uses not only Theorem 5.43 but also Theorem 5.42. The same comment applies to the first sentence of the proof of Theorem 5.48 on the next page.

Page 167

In the line following Definition 5.47, for “absolute integrable” read “absolutely integrable”.

In line 2 of the proof of Theorem 5.48, the closed interval \([a,b]\) should be the open interval \((a,b)\).

In Example 5.49, the notion of conditional integrability on a closed interval has not actually been defined, but you can probably guess the meaning by analogy with the last paragraph on page 164.

Page 168

Strictly speaking, the calculation in the first sentence makes sense only when \(n\ge 2\), rather than “for each \(n\in\mathbf{N}\)”.

Page 169, Exercise 5.4.4

The answer in the back of the book uses the words “converges” and “diverges”, which have not been defined for integrals. If a locally integrable function is improperly integrable, then the improper integral is said to converge; otherwise the integral is said to diverge.

The phrasing of part b is not entirely compatible with Definition 5.38, according to which the notion of improper integrability arises only for functions taking real values (not extended real values). The intent of the question is to ask whether \(f\) is improperly on each of the intervals \([-1,0)\) and \((0,1]\).

Page 170, Exercise 5.4.10

The hint in the back of the book is unnecessarily elaborate, for \(\sin x \ge 2x/\pi\) on the whole interval \([0,\pi/2]\).

Page 172

In the first displayed equation, the sum equals \(n\) only when \(n\) is odd; when \(n\) is even, the sum equals \(n-1\). (The conclusion that the sum tends to \(\infty\) is unaffected.)

Page 172

Strictly speaking, Definition 5.55 does not define the value of \(\Phi\) at the left-hand endpoint \(a\), for the word “partition” has not been defined for degenerate intervals. The only sensible definition is to set \(\Phi(a)\) equal to \(0\).

In Theorem 5.56iii, the inequality is actually an equality, and you should be able to see why.

Page 173

In the proof of Theorem 5.56, part ii, rather than citing the Monotone Property of Suprema, one could invoke the immediately preceding displayed equation, according to which \(\Phi(y)-\Phi(x)\ge 0\) when \(x\lt y\).

Page 174

The hint in the back of the book for Exercise 5.5.1c should say “large” rather than “greatest”. The extreme values of \(\phi\) are not available in closed form, for they occur at points where \(x^2\) is slightly displaced from the reciprocal of an (odd) integral multiple of \(\pi/2\) (as you will see if you compute the points where the derivative of \(\phi\) is equal to zero).

In Exercise 5.5.2, the calculations in parts a and b are inadequate to prove part c. The reason is similar to the situation in Exercise 5.5.1: namely, the critical points are not precisely at reciprocals of integral multiples of \(\pi/2\). A better approach to this exercise is to apply Exercise 5.5.5a.

Exercise 5.5.5b is supposed to illustrate that a function of bounded variation can have an unbounded derivative. The example loses force because the derivative fails to exist at the midpoint of the indicated interval. The same function restricted to the interval \([0,1]\) would be a better example.

Page 177

The first paragraph of the proof of Theorem 5.61 uses an unstated corollary of Remark 5.60. As stated, Remark 5.60 applies only to chords that have a common endpoint. But it follows easily that if \(c_1\lt d_1 \lt c_2 \lt d_2\), then the slope of the chord through the points \( (c_1,f(c_1))\) and \((d_1, f(d_1))\) is less than or equal to the slope of the chord through the points \((c_2,f(c_2))\) and \((d_2,f(d_2))\).

Page 178

In the displayed equation in the proof of Theorem 5.62, the interval \((x_0,b)\) should be \((x_0,d)\).

Page 179

In the proof of part ii of Theorem 5.63, the interval \((a,b)\) should be \([0,\infty)\). The number \(x_1\) should have the property that \(x_0\lt x_1 \lt x_0+\delta\), where \(\delta\) is the number that arises from the definition of proper maximum stated just before the theorem. (The property is defined to be local.)

Page 181

The asterisks on Corollary 5.68 and Theorem 5.69 seem superfluous, for the whole of Section 5.6 is starred. A similar comment applies to Corollary 5.70 on page 182 and Exercise 5.6.8 on page 183.

Notice that the second sentence from the end of the proof of Corollary 5.68 does not really use the continuity of both derivatives. The continuity of \(D_R\) is needed, but \(D_L\) is evaluated at a stationary point, so the continuity of \(D_L\) is not used.

Harold P. Boas