$f:V\rightarrow \mathbb{R}$ with $V\subset \mathbb{R}^n$ is called differentiable if all the partial differentials exist.
I have a confusion about the definition of a differentiable function $\phi:U\rightarrow \mathbb{R^n}$. The definition on my book is as follows:
$\phi:U\rightarrow \mathbb{R^n}$ is called differentiable if any pullback $f\circ\phi$ is differentiable for any differentiable $f$ defined on the range of $\phi$? (Here, I stick for $U$ open in $\mathbb{R}^m$)
The problem is that nothing is told about the domain of $f$. Are the following definitions equivalent or do they imply each other in any way?:
Def-1: $f$ above has a domain $\phi(U)$.
Def-2: $f$ above has a domain $W\supset \phi(U)$ for any open $W$.
Def-3: $f$ above has a domain $\mathbb{R}^n$.