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