Generalized Approach to Differentiability

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of the domain of function or for functions with an open domain. This approach leads to the fact that differentiability has usually been considered only for functions with an open domain in ${\mathbb{R}}^n$, which severely limits the possibility of applying the potential techniques and tools of differential calculus to a broader class of functions. Although there is a great need for generalization of the notion of differentiability of a function in various problems of mathematical analysis and other mathematical branches, the notion of differentiability of a function at the non-interior points of its domain has almost not been considered or successfully defined. In this paper, we have generalized the differentiability of scalar and vector functions of several variables by defining it at non-interior points of the domain of the function, which include not only boundary points but also all points at which the notion of linearization is meaningful (points admitting nbd rays). This generalization allows applications in all areas where standard differentiability can be applied. With this generalized approach to differentiability, some unexpected phenomena may occur, such as a function discontinuity at a point where a function is differentiable, the non-uniqueness of differentials… However, if one reduces this theory only to points with some special properties (points admitting a linearization space with dimension equal to the dimension of the ambient Euclidean space of the domain and admitting a raylike neighborhood, which includes the interior points of a domain), then all properties and theorems belonging to the known theory of differentiability remain valid in this extended theory. For generalized differentiability, the corresponding calculus (differentiation techniques) is also provided by matrices—representatives of differentials at points. In this calculus the role of partial derivatives (which in general cannot exist for differentiable functions at some points) is taken by directional derivatives.