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1. Exercise 2.9.1 part (a), p.49 ...that should be square root of beta. [Thanks to Trent Vaughn]
2. Example 7.29, p.302. The Dini dervatives there are, in fact, computed at an irrational x. [Thanks to Dan Kaneswke]
3. Exercise 2.14.6. The sequence should be assumed to be positive. [Thanks to Jared Bunn]
4. In Section 11.3 describing limit points in Rn, a "limit point" is defined as a point in the set E. It need not be in the set.
This
would contradict later on , Theorem 11.18, that says a set is closed
if it contains all its limit points, which it always does by the
previous incorrect definition. [Thanks to Brian Busemeyer]
5. In Exercise 8.10.1, the upper and lower sums are both defined with inf, but the lower sum should be defined with a sup.
[Thanks to Brian Busemeyer]
6. In Exercise 7.6.2
you require alpha to be positive, but I believe you must also require
beta to be positive. Consider the case when alpha = 7.22 and beta =
-6.25. In this case there are actually two solutions. [Thanks to Brian Busemeyer]
7. Kirsten and Lauren mentioned that there was a typographical error in Exercise 7.6.2:
they found a counter-example to the statement. And, they deduced that
the problem was probably meant to have $beta >0$ rather than
$alpha >0$.
[Thanks to Kirsten Hogenson and Lauren Herrmann, who were
students of Professor Bruce Deardon in 2010 and spotted numerous errors
in the dripped version, most that apply to this version.]
8. Lauren Herrmann, pointed out that for Exercise 7.3.21 the corresponding Note (177) on page the m and n in
the exponent have been reversed. That is, the Power Rule for Rational
Exponents Theorem (7.14,) has $f(x)=x^frac{m}{n}$ while Note,
essentially, uses $x^frac{n}{m}$, slightly confusing the issue.
[Lauren Herrmann was
a students of Professor Bruce Deardon in 2010 and spotted numerous
errors in the dripped version, some that apply to this version.]
9. Exercise 2.4.13. From John Simpson (student at UCSB) "It states that the variable "M" is an integer, but {s_n} is a sequence, which we have defined for only n
in the naturals. Thus, by making M negative, we arrive at values which
are not, strictly, well defined. Should M be a nonnegative integer?"
YES.
[Thanks to Professor C. Akemann for forwarding this one.]
10. Exercise 5.6.14. Thanks to Professor Philip D. Loewen (Univ. of British Columbia)
who has pointed out that Exercise 5.6.14 is quite incorrect.
Here is a suggested replacement:
Exercise 5.6.14[revised]
Part A. Show that a set E is compact if and only if
every continuous function on E is bounded.
Part B. Characterize those sets with the property that
every continuous function on E is uniformly continuous on E.
Spoiler Alert: Part A is easy enough without a hint. For Part B the set E should be either a compact set; or else a compact set
together (possibly) with an increasing sequence x1 < x2 < x3 < ... for which xn+1 - xn > c for some positive c, also together (possibly) with
a decreasing sequence y1 >y2 > y3> ... for which yn - yn+1 > c. Interested readers can consult this MONTHLY paper for a full account of this problem in the general settting of a metric space.
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02/14/2012
