Definition:Differentiable Mapping/Real-Valued Function/Open Set

Definition
Let $\mathbb X$ be an open subset of $\R^n$.

Let $f: \mathbb X \to \R$ be a real-valued function.

Then $f$ is differentiable in the open set $\mathbb X$ :
 * $(1): \quad f$ is differentiable at each $x$ in $\mathbb X$
 * $(2): \quad$ the partial derivatives of $f$ are continuous mappings from $\mathbb X$ to $\R$.

This can be denoted:
 * $f \in \mathcal C^1 \left({\mathbb X, \R}\right)$

Also see

 * Characterization of Differentiability for clarification of this definition.
 * Definition:Differentiability Class for insight into the notation $\mathcal C^1 \left({\mathbb X, \R}\right)$