Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 5Differentials and differentiability

1 INTRODUCTION

Let us consider a function f : S → ℝ^m, defined on a set S in ℝⁿ with values in ℝ^m. If m = 1, the function is called real‐valued (and we shall use ϕ instead of f to emphasize this); if m ≥ 2, f is called a vector function. Examples of vector functions are

Note that m may be larger or smaller or equal to n. In the first example, n = 1, m = 2, in the second n = 2, m = 3, and in the third n = 3, m = 2.

In this chapter, we extend the one‐dimensional theory of differential calculus (concerning real‐valued functions ϕ : ℝ → ℝ) to functions from ℝⁿ to ℝ^m. The extension from real‐valued functions of one variable to real‐valued functions of several variables is far more significant than the extension from real‐valued functions to vector functions. Indeed, for most purposes a vector function can be viewed as a vector of m real‐valued functions. Yet, as we shall see shortly, there are good reasons to study vector functions.

Throughout this chapter, and indeed, throughout this book, we shall emphasize the fundamental idea of a differential rather than that of a derivative as this has large practical and theoretical advantages.

2 CONTINUITY

We first review the concept of continuity. Intuitively, a function f is continuous at a point c if f(x) can be made arbitrarily close to f(c) by taking x sufficiently close to c; in other words, ...