# Definition If $f(x_1, x_2, \ldots, x_n)$ is a function (i.e. [[Map|map]]) defined on some [[Open Set|open region]] $S$ of $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]), then it is said to be *differentiable of class $C^k$* if all its partial derivatives of order less than or equal to $k$ exist and are [[Continuous Map|continuous]] functions on $S$.