The Basic Concepts of Multivariable Functions

The jump from a function of one variable to a function of several variables is not a small one. On the one hand the domain changes from an interval to a region, and on the other hand every concept built on top of the domain has to be rebuilt: a limit now has to hold along infinitely many paths rather than just two directions, and continuity acquires new failure modes (a whole curve of discontinuities rather than a point).

This chapter establishes the vocabulary. It begins with the definition of a multivariable function, the domain and the geometric picture of a surface over a region, together with the level curves and level surfaces that make such a surface readable. It then turns to the limit, where the path method and the iterated limit give two complementary ways of deciding existence or non-existence, and finally to continuity, the properties of continuous functions on a bounded closed region, and the distinction between continuity and uniform continuity.

The rest of the course rests on this foundation. Once you are comfortable with the double limit, the partial derivatives and the total differential that follow are no longer a new idea but the same idea applied in several directions at once.