Differentiability of the $n$-Variable Function Deduced by the Differentiability of the $n-1$-Variable Function

TL;DR

This paper establishes weaker sufficient conditions for the differentiability of an n-variable real-valued function based on the differentiability of its (n-1)-variable counterpart, advancing the theoretical understanding of multivariable calculus.

Contribution

It introduces new, less restrictive conditions for differentiability that depend on the differentiability of lower-dimensional functions, extending classical results.

Findings

Derived weaker sufficient conditions for n-variable differentiability

Connected differentiability properties between n-variable and (n-1)-variable functions

Enhanced theoretical framework for multivariable function analysis

Abstract

In this paper, some sufficient conditions for the differentiability of the $n$-variable real-valued function are obtained, which are given based on the differentiability of the $n-1$-variable real-valued function and are weaker than classical conditions.