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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.

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  • $\begingroup$ (I) A singelton $A = \{* \} $ is a $0$-manifold, and $f$ is differentiable with $df(* ) \colon T_* \{* \} = \{0\} \to T_{f(* )}\mathbb R = \mathbb R$ the zero map. (II) There is only one such function and it depends on the definition of differentiable mainfold if $\emptyset$ is one, I suppose. (III) $\lim_{\substack{t \to x\\ t \in P}}$ looks common to me. $\endgroup$ – martini Nov 13 '12 at 14:24
  • $\begingroup$ @martini Hmm.. Would you explain it a little easier? I don't know any concept of manifold. And how do i write that in LaTeX? $\lim_{t\to x}{t\in P}$ $\endgroup$ – Katlus Nov 13 '12 at 14:31
  • $\begingroup$ In LaTeX write \lim_{\substack{t \to x\\ t \in P}} ... regarding your first two question: How do you define being differentiable on a set $A$ with $|A| \le 1$, then? $\endgroup$ – martini Nov 13 '12 at 14:35
  • $\begingroup$ @martini See my edited post. I thought the definition is that (in my post) so it cannot be defined such function for a singleton, i wasn't sure if differentiability can be defined for a singleton. $\endgroup$ – Katlus Nov 13 '12 at 14:41
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    $\begingroup$ @martini I didn't like the way differentiability can be defined on a real function whose domain is an infinite connected set in $\mathbb{R}$, so i wanted a generalization. However, it seems like i cannot understand smooth manifold concept on my level.. Anyway many thanks! $\endgroup$ – Katlus Nov 13 '12 at 14:51

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