ABSTRACT
This chapter presents the notion of a function between several variables. It defines domain, codomain, level sets and graph of a function of several variables, and also discusses quadrics and curves. As a rule of thumb, when establishing the domain of definition of a function one has to take into account that division by zero, taking square roots and logarithms of negative numbers are not allowed. Under the mere condition of continuity, the range of a function is called a surface. The fact that all elementary functions, that is, polynomial, rational, trigonometric, exponential and logarithmic functions in several variables are continuous. The vertical line test that determines whether a shape in R2 is the graph of a function works further for functions of two variables. Indeed, if f(x,y) is a function of two variables, then any vertical line must intersect the traces, and thus the whole surface z=f(x,y), in at most one point.
