Notation question, $x=1,2,3...$ or $x\in \mathbb N$? On my recent exam, my professor wrote a note saying that the following notation shouldn't be used when discussing domains, and I'm not sure why it matters. So, my question:
I'll use a specific example, but I'm speaking generally when it comes to notation. When given a function of the form:
$$f(x) = \cases{ g(x)  & if $x=1,2,3,...$  \\  h(x) & if $ x= -1, -2, -3, ..$}    $$
Is it incorrect to write it as: 
$$f(x) = \cases{ g(x)  & if $x\in \mathbb N$  \\  h(x) & if $ x\in \mathbb Z<0$}    $$
I've always been curious why this notation isn't used in textbooks, and my professors comment on my exam peaked my curiosity enough to post it here.

EDIT: I should mention I come from a physics and engineering background, not mathematics.

Thank you!
 A: Ignoring the question of zero's status w.r.t. $\mathbb{N}$: There is some flexibility and looseness in notation that most people will allow, but $$x \in \mathbb{Z} \lt 0$$ just doesn't work (for me).
Here's why: people often write $$ a R_1 b R_2 c$$ as an abbreviation of $$a R_1 b \text{ and } b R_2 c,$$ where $a$, $b$, and $c$ are variables or constants and $R_1$ and $R_2$ are binary relation. So you can write $a \lt b \lt c$ to mean $a\lt b$ and $b \lt c$, or even $0 \lt x \in \mathbb{R}$ to mean $0 \lt x$ and $x \in \mathbb{R}$, but your expression leaves me with $\mathbb{R} \lt 0$ for the second half, which is either confusing or nonsensical. 
I guess you could stay within the limits I've prescribed and write it as $$ 0 \gt x \in \mathbb{N},$$ which, if I came across, I would think "Hmm, that's a little weird, never seen it before" but it would be clear to me what you meant. Math is a language and there are lots of sentences that are grammatical and you could write, but people generally don't write. So using them adds a bit of unnecessary confusion that it's probably better to avoid if you can.
Regardless, I find your first way of writing it more straight forward and would prefer it. It also lets you avoid all the hubbub about $0 \in \mathbb{N}$ versus $0 \notin \mathbb{N}$ that you see in the comments. 
