Definition:Continuously Differentiable/Real-Valued Function

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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 $U$.