I am reading "Calculus on Manifolds" by Michael Spivak.
The following lemma is on p.35 in this book.
Lemma 2-10 : Let $A \subset \mathbb{R}^n$ be a rectangle and let $f : A \to \mathbb{R}^n$ be continuously differentiable. If there is a number $M$ such that
$| D_j f^i (x) | \leq M$ for all $x$ in the interior of $A$, then
$$
|f(x)-f(y)| \leq n^2 M |x-y|
$$
for all $x,y \in A$.
If $A$ is an open rectangle, then "$| D_j f^i (x) | \leq M$ for all $x$ in the interior of $A$" is strange because the interior of $A$ is equal to $A$.
So, if $A$ is an open rectangle, "$| D_j f^i (x) | \leq M$ for all $x$ in $A$" is natural.
So, I think $A$ is a closed rectangle.
But if $A$ is a closed rectangle, then the author didn't need to write the following since this automatically holds:
If there is a number $M$ such that
$| D_j f^i (x) | \leq M$ for all $x$ in the interior of $A$
Is $A$ open or closed?
Since this lemma is used in the proof of the inverse function theorem, so I think it is best to read the proof of the inverse function theorem first.
I think I can decide $A$ is open or closed.
Best Answer
The Lemma, in general, only assumes the inequality and not the openness or closedness of the rectangle, just that it is some rectangle. As you've pointed out, if $A$ is a closed rectangle, then the inequality he assumes is automatically satisfied. When the rectangle is open, the assumption is not redundant (e.g. $f(x) = \sqrt{x}$ and $A = (0,1)$).