# Basic Analysis II: Introduction to Real Analysis, Volume II: Errata

Every book (no matter how much you paid for it) has errors and typos, especially text that is new in a given edition. I try to be as transparent as possible about any errors found, and I try to fix them as quickly as possible. Please do let me know if you find any errors or typos, so that they may be fixed. A free book relies on readers sending in corrections.

This page lists errors in the various editions. Nonmathematical typos, misspellings, and grammar or style problems are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions may suffer from recently discovered errata as well.

June 10th 2020 edition, Version 2.3 (edition 2, 3rd update):

1. Definition 9.2.1, page 66: The definition of "a simple path" is still not quite right for non-closed simple paths. We should require that if $\gamma$ is not one-to-one on the entire interval then it should be equal at the endpoints (no biting itself in the middle): "A path $\gamma \colon [a,b] \to {\mathbb{R}}^n$ is simple if either 1) $\gamma$ is one-to-one, or 2) $\gamma|_{[a,b)}$ is one-to-one and $\gamma(a)=\gamma(b)$." The wording of Exercise 9.3.7 already assumed this correct definition.
2. In Definition 9.2.5, page 68, for a piecewise smooth $h$ the fact that all one-sided limits of $h'$ are nonzero is forgotten, although it is correctly explained below, and correct in the definition for a smooth reparametrization.
3. Exercise 9.2.14, page 76: The explanation of what not being one-to-one except on a finite set was misleading it was only a consequence not an if and only if (to be an if and only if it was missing that the preimages are always finite). It is better to just state directly what being one-to-one except on a finite set means. Same issue is on page 71 where the exercise is referred.
4. Exercise 10.5.3, page 115: The function defined on $U$ should clearly be named something else than $f$. Say $\varphi$.
5. Exercise 10.5.4, page 116: Definition of $W$ should subtract the complement of $U$, not $U$. Actually an easier way to write what $W$ should be is $\bigl( U \times (0,2) \bigr) \cup \bigl( (0,1) \times (1,2) \bigr)$.
6. Exercise 10.6.4, page 120: In part c), the first $dy$ should be a $dx$, that is, the integral should be $\displaystyle \int_{\partial B(p,r)} \frac{-y}{2} ~ dx + \frac{x}{2} ~ dy$.
7. In the proof of 10.7.2, throughout the proof, $(1+4\sqrt{n}\, \epsilon)$ should be ${(1+4\sqrt{n}\, \epsilon)}^n$ and in the long estimate on page 123, the three instances of $(1+4\sqrt{n}\, \epsilon)$ should be $\frac{1}{{(1+4\sqrt{n}\, \epsilon)}^n}.$
8. The statement of Theorem 11.3.1 is confusingly worded. The three items are three distinct possibilities.
9. In lemmas 11.5.1 and 11.5.4 and theorem 11.5.5 (FTA), the hypothesis that the polynomials are nonconstant is missing.
10. In the caption to figure 11.7 it says the vertical axis is from $-e^6$ to $e^6$, but it should say $-e^4$ to $e^4$. The approximate value $54.6$ is correct.

May 15th 2019 edition, Version 2.2 (edition 2, 2nd update):

1. On page 35, towards the end of the proof of the chain rule, it says that ${\|f(p+h)-f(p)-Ah\|}{\|h\|}$ is bounded, when it should be, of course, $\frac{\|f(p+h)-f(p)-Ah\|}{\|h\|}$.
2. On bottom of page 47, defining $\varphi_y$, the domain of $\varphi_y$ should be $V$ not $C$.
3. Definition 9.2.1, page 67: definition of "a simple path" is not quite right, it should be that $\gamma$ is one-to-one on $[a,b)$, not just $(a,b)$. The error is then carried over in the couple of places where "simple" is used in the examples and exercises: Example 9.2.2, discussion on page 71, exercise 9.2.13.
4. On page 145, end of proof of Theorem 11.3.9. We have only proved that the set of cluster points of $E$ is open (we claimed that we proved that $E$ is open), but the set of cluster points is also closed and so equal to $U$ anyway.
5. On page 157, when we say we enumerate $X$ in the proof of Proposition 11.6.5, it makes it seem like $X$ is necessarily infinite, which is the important case, but the proof should work even for finite $X$.
6. On page 160, Exercise 11.6.1 only asks about why the sequence does not converge uniformly, it should ask about a subsequence. The solution is essentially the same, and we are asking about why Arzelà-Ascoli does not apply so we must ask about a subsequence.

October 11th 2018 edition, Version 2.1 (edition 2, 1st update):

1. On page 102, middle of the page in the estimate for $\sum r_k^n$, the right hand side should be ${(4\sqrt{n})}^n \epsilon$.

May 7th 2018 edition, Version 2.0 (edition 2, 0th update):

1. On top of page 35 (right after Theorem 8.3.7, Chain Rule) it says $(f \circ g)'$ when it ought to be $(g \circ f)'$. Thanks to Trevor Fancher.

March 21st 2017 edition, Version 1.0 (edition 1, 0th update):

1. On page 10, when solving for $x_j$, the coefficients must be negative, that is, $x_j = \frac{-a_1}{a_j} x_1 + \cdots + \frac{-a_{j-1}}{a_j} x_{j-1} + \frac{-a_{j+1}}{a_j} x_{j+1} + \cdots + \frac{-a_k}{a_k} x_k .$
2. In Exercise 8.2.13, part a), it should say "for all $c\in {\mathbb{R}}^n$" instead of just "for"
3. On page 76, in the definition of the Darboux sums, $P$ is a partition of $R$.
4. In Exercise 8.3.8, the hypothesis should say $\nabla f(0,0) = (0,1)$. Thanks to Trevor Fancher.
5. In the Inverse function theorem, Theorem 8.5.1, the $U$ should be an open set.
6. In Exercise 8.5.6, the hypothesis should be that the first component of $\nabla f(t)$ is not zero.
7. In the proof of Theorem 9.1.1, an offhand remark is made about replacing $h$ with $\frac{1}{n}$. That wouldn't work, we need an arbitrary sequence $\{ h_n \}$ converging to zero.
8. Exercise 9.1.7 part c) is impossible (not true). Removing part c makes the exercise quite uninteresting so it will be replaced with a different exercise (demonstrating the same issue) in the next edition.
9. In definition of smooth path, we really assume that derivative of a function of one variable has been defined, although this was only properly defined for the components. Thanks to Trevor Fancher for pointing this out.
10. In the proof of Proposition 9.2.6, the restriction of $h$ in the first paragraph is to $[r_{j-1},r_j]$, not $[t_{j-1},t_j]$. In the second paragraph, the $\varphi$ should be a $\gamma$.
11. In Definition 9.2.7, the functions $f_1,\ldots,f_n$ should be named $\omega_1,\ldots,\omega_n$ as those are used in the definition.
12. In Definition 9.2.9, when talking about the partition being minimal we mean $t_1,t_2,\ldots,t_{k-1}$ not $t_2,t_3,\ldots,t_{k-1}$. Also in the definition, in the displayed equation for the piecewise smooth definition, the last interval is marked as $[t_{n-1},t_n]$ when it should of course be $[t_{k-1},t_k]$.
13. Exercise 9.3.6 should require that $U_1 \cap U_2$ is path connected, we don't want to have to use another exercise here.
14. Exercise 9.3.9 is for an open set $U$, while this is tacitly implied as we have not defined path-connected for other sets, it should be explicit.
15. In proof of theorem 10.1.15, the expression $M_i-m_i$ should be $M_k-m_k$
16. In the proof of Proposition 10.3.2, the proof of the reverse direction does not make sense. There is no need estimate $r_j^n$ with $r_j$.
17. In the construction of Cantor set, $j$ and $n$ are used interchangeably. It should all just be $n$.