Definition:Continuously Differentiable/Real-Valued Function
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Definition
In an Open Set
Let $U$ be an open subset of $\R^n$.
Let $f: U \to \R$ be a real-valued function.
Then $f$ is continuously differentiable in the open set $U$ if and only if:
- $(1): \quad f$ is differentiable in $U$.
- $(2): \quad$ the partial derivatives of $f$ are continuous in $U$.