I am trying to self-study the textbook "An Introduction to Convexity, Optimization, and Algorithms" by Walaa M. Moursi. In part 3.1 she states that:
$f$ is called proper if $dom(f) \neq \emptyset$ and every $x \in X$ satisfies $-\infty < f(x)$.
Keep in mind that, in this book, $X$ stands for Euclidean space. Also, she defines (effective) domain to be:
$dom(f) := \{x \in X \mid f(x) < +\infty \}$
Then she gives an example of proper functions and says:
Any function $f: X \rightarrow \mathbb{R}$ is proper.
And this is where I don't understand what is going on. For example, define $f := -\frac{1}{\lvert x\rvert}$, then by her definition $dom(f) = \mathbb{R}$, but $f(0) = -\infty$, so $f$ is clearly not proper. Can anyone please explain?