Let $D\subset\mathbb R$ and let $f:D\to\mathbb R$ be a function $f$ is called continuously differentiable, if its derivative $f':D\to\mathbb R$ is continuous. Correspondingly, if $f$ has a derivative of order $n$ $f^{(n)}$ on $D$ and if $f^{(n)}$ is continous, the function $f$ is called $n$ times continuously differentiable.
Lemmas: 1 2
Proofs: 3 4 5
Theorems: 6 7