I am reading from the book Multidimensional real analysis vol I by Duistermaat and Kolk and am trying to understand the following theorem from it:
Theorem 1.3.2. In the notation of Definition 1.3.1 we have $\lim_{x\to a}f(x)=b$ if and only if for every neighborhood $V$ of $b$ in $\mathbb R^p$ the inverse image $f^{-1}(V)$ is a neighborhood of $a$ in $A$.
Definition 1.3.1 is as follows:
Definition 1.3.1. Let $A \subset \mathbb{R}^n$ and let $a \in \overline{A}$; let $f \colon A \to \mathbb{R}^p$ and let $b \in \mathbb{R}^p$. Then the mapping $f$ is said to have a limit $b$ at $a$, whith notation $\lim_{x \to a} f(x) = b$, if for every $\epsilon > 0$ there exists $\delta > 0$ satisfying $$ x \in A \quad \text{and} \quad \| x - a \| < \delta \quad\implies\quad \| f(x) - b \| < \epsilon. $$
(Original image here.)
My question is: It is clear that the theorem implies that $\lim_{x\to a}f(x)=b\implies a\in A$, as $f^{-1}(V)$ is a neighborhood of $a$ in $A$ implies $a\in A$, which is not always true. I believe the theorem as stated is incorrect and this is a mistake in the book. How may it be rectified?
The definition of neighborhood in the book is as follows. A set $U$ is a neighborhood of $x$ in $A$ if $U$ contains an open set in $A$ which contains $x$.
