On page 27 of Lee's Introduction to Smooth Manifolds he states
a map from an arbitrary subset $A\subseteq\mathbb{R}^n$ to $\mathbb{R}^n$ is said to be smooth if in a neighborhood of each point of $A$ it admits an extension to a smooth map defined on an open subset of $\mathbb{R}^n$ (see Appendix C, p. 645).
The section which the author is probably pointing to in Appendix C reads:
If $A\subseteq\mathbb{R}^n$ is an arbitrary subset, a function $F:A\to\mathbb{R}^m$ is said to be smooth on $A$ if it admits a smooth extension to an open neighborhood of each point, or more precisely, if for every $x\in A$, there exist an open subset $U_x\subseteq R^n$ containing $x$ and a smooth function $\hat{F}:U_x\to\mathbb{R}^m$ that agrees with $F$ on $U_x\cap A$.
What is meant by the above definition?
My particular issue is the following: if $U_x$ need not be contained in $A$, then the value of the derivative will not be well-defined for some sets e.g. $$f:\{0\}\to\mathbb{R}$$ may be extended in multiple ways regardless of the value of $f(0)$.
If, on the other hand, $U_a$ is contained in $A$, then differentiability remains only defined for open sets, not arbitrary ones.
Thus neither case seems to portray what the author is saying.