Continous Differentiability

Continuous Differentiability

Statements

Definition. (@Boo75, pp. 22) Let $U\subset {\mathbb{R}}^n$ be an open set. Let $r\in {\mathbb{N}}_{>0}$, a function $f:U\to \mathbb{R}$ is said to be $r-$continuously differentiable or of class ${\mathcal{C}}^r$ on $U$, denoted as ${\mathcal{C}}^r(U,\mathbb{R})$, if it has continuous derivatives of order 1, 2, …, $r$ on $U$. If $r=\infty$, then is said to be smooth.