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

Also see

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differentiable
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differentiable