The Various Definitions of Multiple Differentiability of a Function f: ℝn→ ℝ

Abstract

Since the 17-th century the concepts of differentiability and multiple differentiability have become fundamental to mathematical analysis. By now we have the generally accepted definition of what a multiply differentiable function f:Rn→R is (in this paper we call it standard). This definition is sufficient to prove some of the key properties of a multiply differentiable function: the Generalized Young’s theorem (a theorem on the independence of partial derivatives of higher orders of the order of differentiation) and Taylor’s theorem with Peano remainder. Another definition of multiple differentiability, actually more general in the sense that it is suitable for the infinite-dimensional case, belongs to Fréchet. It turns out, that the standard definition and the Fréchet definition are equivalent for functions f:Rn→R. In this paper we introduce a definition (which we call weak) of multiple differentiability of a function f:Rn→R, which is not equivalent to the above-mentioned definitions and is in fact more general, but at the same time is sufficient enough to prove the Generalized Young’s and Taylor’s theorems.