Class C^k map

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).

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 $x=0$.