  In this paper, several differentiability criteria for real functions of
multiple variables in n-dimensional Euclidean space are considered. Simple and
easy-to-use Cauchy-like criterion is formulated and proven. Relaxed sufficient
conditions for differentiability that do not require continuity of all partial
derivatives are suggested. Generalization of the Cauchy-like criterion for
functions on cross products of normed vector spaces (not necessarily Banach
spaces) is discussed. The results of this study can be used in systems
analysis, linear programming, optimization methods, functional analysis,
topology and convex analysis.
