Class C^k map - Knowlpedia

For a nonnegative integer $k$ , a class $C^k$ map is a function $F:U\to\mathbb{R}^m$ defined on an open set $U\subseteq\mathbb{R}^n$ such that all partial derivatives of $F$ of total order at most $k$ exist on $U$ and are continuous on $U$ (with order $0$ meaning $F$ itself).

A map is class $C^\infty$ , or smooth, if it is class $C^k$ for every nonnegative integer $k$ .

Derivative descriptions

When $k=1$ , this is the standard “continuously differentiable” hypothesis in multivariable calculus: the derivative is encoded by the Jacobian matrix, and $C^1$ regularity is closely aligned with having a continuous Fréchet derivative. Higher regularity interacts with mixed derivatives via results like the Schwarz–Clairaut theorem.

Examples

The map $F(x,y)=(x^2+y,\;xy)$ is class $C^\infty$ on $\mathbb{R}^2$ .
The map $F(x,y)=(|x|,\;y)$ is class $C^0$ on $\mathbb{R}^2$ but not class $C^1$ along the line $x=0$ .