This is in the context of defining the derivative. Here is the full quote from Calculus on Manifolds, pg 17:
The definition of $Df(a)$ could be made if $f$ were defined only in some open set containing $a$. Considering only functions defined on $\mathbb{R}^n$ streamlines the statement of theorems and produces no real loss of generality.
Why is it no loss of generality? What is the formal statement that is being made?
The corresponding notion on $\Bbb R$ is something like this:
"We could define differentiability for a function on an open interval, but we'll be defining it for functions on $\Bbb R$ without loss of generality. The definition involves limits, for example, and so rather than writing something like
$$ \lim_{x \in (a, b), x \to c} \ldots $$ we can simply write $$ \lim_{x \to c} \ldots . $$
Why? Because for every function $f$ on the interval $(a, b)$, we can define a function $\hat{f}$ on $\Bbb R$ via $$ \hat{f}(x) = \begin{cases} f(x) & a \le x \le b \\ 0 & \text{otherwise} \end{cases}. $$
And for any $c \in (a, b)$, the limits of $f$ and $\hat{f}$, as we approach $c$ will be identical, since limits involve only local properties."