Let $A$ be a singleton.
(I)Does there exist $f:A\rightarrow \mathbb{R}$ differentiable on $A$?
(II)Likewise, is every function $f:\emptyset \rightarrow \mathbb{R}$ differentiable on $\emptyset$?
I think the first one is false but the second one is true since "$\forall x\in \emptyset, \exists g:\emptyset \setminus \{x\} \rightarrow \mathbb{R}:t \mapsto \frac{f(t)-f(x)}{t-x}$" is vacuously true. Am i correct?
Additional Question: is there any notation for a limit with it's domain? For example, let $f:A\rightarrow B$ be a function and $P\subset A$ and $x$ be a limit point of $P$. Then $\lim_{t\to x} f\upharpoonright P (t)$may differ from $\lim_{t\to x} f(t)$
I think i first need to know what is the precise definition of differentiation, since i thought a real function $f$ is differentiable at $x$ in its domain $A$ iff $\lim_{\substack {t\to x \\t\in A\setminus \{x\}}} \frac{f(t)-f(x)}{t-x}$. But now it seems like generally it's not the definition after i saw martini's comment.