1
$\begingroup$

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.

  • 1
    @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!2012-11-13

0 Answers 0