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.

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

No known errata.

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 $n$.  