In a class I took on Analysis in Several Real Variables, on the first day the lecturer wrote on the blackboard
Definition A function of the form
$\begin{align*} \mathbb{R}^n \supset V &\overset{f}{\to} \mathbb{R} \\(x_1,\ldots,x_n) = x &\mapsto f(x) = f(x_1,\ldots,x_n) \end{align*}$
is called a real valued function of n independent real variables if $V$ is open in $\mathbb{R}^n$
He then stated to the class "Note: independent is equivalent to on open set".
I duly noted this in my book suffixed with a sarcastic "(apparently)" since at the time it was one of those things which made absolutely no intuitive sense to me.
Looking over my notes now, at the end of the year, this still makes little sense, can someone help clarify it to me?