Definition. $\mathscr{C}'$-mapping [br920]

A differentiable mapping $\mathbf{f}$ of an open set $E \subset \mathbb{R}^{n}$ into $\mathbb{R}^{m}$ is said to be continuously differentiable in $E$ if $\mathbf{f}'$ is a continuous mapping of $E$ into $L(\mathbb{R}^{n}, \mathbb{R}^{m})$.

More explicitly, it is required that to every $\mathbf{x} \in E$ and to every $\epsilon > 0$ corresponds a $\delta > 0$ such that $$ \lVert \mathbf{f}'(\mathbf{y}) - \mathbf{f}'(\mathbf{x}) \rVert < \epsilon $$ if $\mathbf{y} \in E$ and $\lvert \mathbf{x} - \mathbf{y} \rvert < \delta$.

If this is so, we also say that $\mathbf{f}$ is a $\mathscr{C}'$-mapping, or that $\mathbf{f} \in \mathscr{C}'(E)$.