Minor Mistake in Computability, Complexity, and Languages?

Question

In the book Computability, Complexity, and Languages (2^nd edition), Martin Davis writes in chapter 1 (Preliminaries), section 2 (Functions):

A partial function on a set $S$ is simply a function whose domain is a subset of $S$. An example of a partial function on $\mathbb{N}$ is given by $g(n) = \sqrt n$, where the domain of $g$ is the set of perfect squares.

So far so simple. But he goes ahead and writes a couple lines later at the end of the section:

We will sometimes refer to the idea of closure. If $S$ is a set and $f$ is a partial function on $S$, then $S$ is closed under $f$ if the range of $f$ is a subset of $S$. For example, $\mathbb{N}$ is closed under $f(n) = n^2$, but is not closed under $h(n) = \sqrt n$ (where $h$ is a total function on $\mathbb{N}$).

So in the first quote $\sqrt n$ on $\mathbb{N}$ is an example for a partial function, whereas in the second quote the same function is an example for a total function.

Both examples seem to contradict each other. Or am I missing something in regard to closures here?