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

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

No known errata.

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 definined 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 interchangably. It should all just be $n$.