Cauchy-like Criterion for Differentiability of Functions of Several Variables

TL;DR

This paper introduces a simple Cauchy-like criterion for the differentiability of multivariable functions, relaxing traditional conditions and extending applicability to functions on cross products of normed spaces, with broad implications in analysis and optimization.

Contribution

It formulates a new, easy-to-apply differentiability criterion that relaxes continuity requirements of partial derivatives and generalizes to functions on various normed spaces.

Findings

Proposes a simple Cauchy-like differentiability criterion.

Provides relaxed conditions not requiring all partial derivatives to be continuous.

Extends the criterion to functions on cross products of normed vector spaces.

Abstract

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.