Continuous Differentiability in the Context of Generalized Approach to Differentiability

Recently, in their paper, the authors generalized the notion of differentiability by defining it for all points of the functional domain (not only interior points) in which the notion of differentiability can be considered meaningful. In this paper, the notion of continuous differentiability is introduced for the differentiable function $f:X\to {\mathbb{R}}^m$ with a not necessarily open domain $X\subseteq {\mathbb{R}}^n$; i.e., the continuity of the mapping $df:X\to Hom\left({\mathbb{R}}^n,{\mathbb{R}}^m\right)$ is considered. In addition to introducing continuous differentiability in the context of this generalized approach to differentiability, its characterization is also given. It is proved that the continuity of derivatives at some not necessarily interior points of the functional domain in the direction of n linearly independent vectors implies (continuous) differentiability.