Definition:Partial Derivative/Real Analysis/Open Set

Let $U\subset\R^n$ be an open set.
Let $f : U \to \R$ be a real-valued function.
Let $f$ be differentiable in $U$.
The $i$th partial derivative (function) of $f$ with respect to $x_i$ is the real-valued function which sends each $x\in U$ to the $i$th partial derivative at $x$.