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 $\mathbb X$ $f$ is differentiable at each point of $\mathbb X$.

Also see

 * Characterization of Differentiability for clarification of this definition.
 * Definition:Continuously Differentiable Vector-Valued Function