How is this definition of differentiability for an arbitrary subset $S$ of $\mathbb{R}^n$ common and used?

Question

In the book of Analysis On Manifold by Munkres, at page 144, in the exercise 3.a, the author defines differentiability for an arbitrary subset $S$ of $\mathbb{R}^n$ as

Let $S$ be an arbitrary subset of $\mathbb{R}^n$; let $x_0 \in S$. We say that $f:S \to \mathbb{R}$ is diff'able at $x_0$, of class $C^r$, provided that there is a a $C^r$ function $g: U\to \mathbb{R}$ defined in a neighbourhood of $U$ of $x_0$ in $\mathbb{R}^n$, such that $g$ agrees with $f$ on the set $U \cap S$.

However, considering the answers given to this question, how is the above definition common and used ?

Answer 1

This is used when defining/working with manifolds with boundary: they are manifolds which are locally diffeomorphic to an open subset of $\mathbb R^{n-1} \times \mathbb R_{\geq 0}$. The transition maps between charts are differentiable in the sense of the definition in your question.

The question you linked to, seems to be asked by someone who is studying real analysis, where we don't care about differentiability on arbitrary subsets, and the definitions explicitly ask for the function to be defined on an open interval around the point in question.