I am reading "Calculus on Manifolds" by Michael Spivak.
I think we need to modify the proof of 1-8 Theorem.
Spivak wrote "Since $f$ is continuous at $a$, we can ensure that $f(x) \in B$, provided we choose $x$ in some sufficiently small rectangle $C$ containing $a$".
I think we need to modify as follows:
"Since $f$ is continuous at $a$, we can ensure that $f(x) \in B$, provided we choose $x$ in the intersection of $A$ and some sufficiently small rectangle $C$ containing $a$"
Am I right or not?
Best Answer
You are right.
In my opinion Spivak is a bit sloppy. He starts at a very elementary level, defines everything ab ovo, but since he does so, he should be more precise. Let me explain what I mean.
On p. 1 he introduces the (Euclidean) norm $\lvert x\rvert$ on $\mathbb R^n$. In exercise 1-5 he introduces the (Euclidean) distance between $x$ and $y$ as the quantity $\lvert x - y \rvert$. This is a metric, but he doesn't mention it.
Open rectangles and open sets $U \subset \mathbb R^n$ are introduced on p. 5, but Spivak does not mention the essential fact that $U$ is open if and only if for each $x \in U$ there exists an $\epsilon > 0$ such that for all $y$ satisfying $\lvert x - y \rvert < \epsilon$ one has $y \in U$. The proof is of course easy. Note that the part after "only if" is the standard definition of an open set.
On p. 11 Spivak defines the limit of a function $f : A \to \mathbb R^m$ at $x \in A$ via the Euclidean distance. This leads to the definition of continuity at $a$. In the proof of Theorem 1-8 he tacitly invokes two facts which he doesn't prove:
There exists $\epsilon > 0$ such that for all $z$ satisfying $\lvert f(a) - z \rvert < \epsilon$ one has $z \in B$. See the above remark concerning the concept of "open set".
By continuity there exists $\delta > 0$ such that for all $y \in A$ such that $\lvert a - x \rvert < \delta$ one has $\lvert f(a) - f(x) \rvert < \epsilon$. Now he says that $f(x) \in B$ provided we choose $x$ in some sufficiently small (of course open) rectangle $C $ containing $a$. This means that $x \in A \cap C$ implies $\lvert a - x \rvert < \delta$. His claim is true, but I think that again there is a gap.
The main shortcoming is that Spivak does not sufficiently precisely address the topology of $\mathbb R^n$. What he should have done is to prove (or at least to mention in an exercise) that
If $x \in \mathbb R^n$ and $\epsilon > 0$, then there exists an open rectangle $R$ containing $x$ such that $y \in R$ implies $\lvert x - y \rvert < \epsilon$.
If $x$ is contained in an open rectangle $R$, then there exists $\epsilon > 0$ such that $\lvert x - y \rvert < \epsilon$ implies $y \in R$.
Anyway, I do not see any reason why Spivak decided to work with open rectangles instead of open balls. An approach based on open balls would be more transparent.