# 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 $U$.