Definition:Continuously Differentiable/Real Function
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Definition
On an Open Interval
Let $I\subset\R$ be an open interval.
Then $f$ is continuously differentiable on $I$ if and only if $f$ is differentiable on $I$ and its derivative is continuous on $I$.
On an Open Set
Let $I \subset \R$ be an open set.
Then $f$ is continuously differentiable on $I$ if and only if $f$ is differentiable on $I$ and its derivative is continuous on $I$.