Cauchy-like Criterion for Differentiability of Functions of Several Variables

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. Used Notations Without loss of generality, we consider differentiability of a real function f( ͢ x) at a point ͢ x = 0, where ͢ x is a vector in n-dimensional Euclidian space: ͢ x ∊ R n (f: U  R, where U is some vicinity of the origin, U ⊂ R n ). Also, without loss of generality, real function f( ͢ x) is assumed to be equal to zero at the origin ͢ x = 0, f(0) = 0. Different vectors ͢ xj are distinguished by different subscript indexes, while their components (space coordinates) are marked with different superscript indexes: ͢ xj ={xj 1 ,.. xj i ,.. xj n }. In our consideration, we use asymptotic notations of big O (ρ) and little o (ρ) that are noted as follows: O (ρ)/ρ is a bounded function when ρ → 0, while [o (ρ)/ρ] → 0 when ρ → 0. Next, we introduce and denote partial values of any function f( ͢ x) as following: f i = f ( ͢ x) = f(x· ͢ ei) = f({0,.. x i ,.. 0}), where ͢ ei is the unit vector for the i-th coordinate axis of the vector space R n . Introduction Any criterion is just the same original definition or statement only rewritten in different terms and notations. Different criteria may not only differ by terms but also by difficulty of their use. In fact, the general opinion is that there is no good or easy-to-use differentiability criterion for multivariable functions. In our paper we show otherwise. Differentiability of multivariable functions is discussed in most of the textbooks on Calculus [1, 2]. In this section, we briefly repeat definitions and general statements with some common examples. Differentiability for a function of n variables is defined in the following way. A function f( ͢ x) is said to be differentiable at the point ͢ x = 0 if there is a finite vector ͢ A ∊ R n such that: f( ͢ x) = ͢ A· ͢ x + o (ρ) , where ρ = | ͢ x |= 2