The following definition for a residue map is given in "Algebraic Curves over a Finite Field" by Hirschfeld, Korchmáros and Torres (page 265):
"Let $\Sigma$ be a field of transcendence degree 1 over $K$. For every place $P$ of $\Sigma$, put $res_P(x) = \begin{cases} a& \text{if } ord_p(x-a)\ge 0 \\ \infty & \text{if } ord_p(x)<0 \end{cases}$ Then the residue map, $x \to res_p$, associated to $p$ is a $K$-morphism of $\Sigma \cup {\infty}$ onto $K \cup {\infty}$."
They go on and use properties of this map, for example linearity. But some things aren't so clear to me with this definition.
How is it well-defined? If $v_p(x-a) \ge 0$ for some $a \in K$, doesn't it follow that $v_p(x-b) \ge 0$ for any $b \in K$ because of the non-archimedian valuation? Shouldn't the $\ge$ be changed to a strict inequality, i.e. $>$?
If we change it to strict inequality, why is guaranteed that such $a$ exists? I agree that such $a$ exists, but not in $K$ - maybe in a larger set.
So, is there a mistake in the book or am I missing something (or maybe both)?