Chapter 4 - Differential Calculus of Real-Valued Functions

Abstract

If D is a subset of Rn, a rule f that assigns a unique real number to each point of D will be called a real valued function on D. The set D on which f is defined is called the domain of f. Usually, a function is defined by a single formula valid over its entire domain D. is the chapter presents an assumption that if f is given by a formula and D is unspecified, then D should be understood to be the largest subset of Rn for which the formula makes sense. The problem of locating extreme values of a function of one variable is an important application of the one-dimensional calculus. This chapter highlights this problem for functions of several variables. It explores limit and continuity, which are fundamental to the calculus of functions of one variable. It generalizes these ideas to real valued functions of several variables and extends them further to vector valued functions. It presents some theorems that allow to build up an extensive collection of continuous functions by addition, subtraction, multiplication, division, and composition of continuous functions. It also describes the derivative of a function of one variable and presents a definition that generalizes the idea of the derivative to functions of which domains are in Rn. The chapter also explains directional and partial derivatives.