Chapter 6

Multi-variable Calculus6.1 Continuous Functions

In what follows, F will denote either R or C. It turns out it is more efficient to not makea distinction. However, the main interest is in R so if you like, you can think R wheneveryou see F.

6.2 Open And Closed SetsEventually, one must consider functions which are defined on subsets of Fn and their prop-erties. The next definition will end up being quite important. It describe a type of subset ofFn with the property that if x is in this set, then so is y whenever y is close enough to x. Inall of this, for x a vector, |x| is given by (x,x)1/2 where this denotes the square root of theinner product of the vector with itself as described earlier. Then the distance between thevectors x and y is defined as |x−y|.

Definition 6.2.1 Let U ⊆ Fn. U is an open set if whenever x ∈U, there exists r > 0 suchthat B(x,r)⊆U. More generally, if U is any subset of Fn, x ∈U is an interior point of U ifthere exists r > 0 such that x ∈ B(x,r)⊆U. In other words U is an open set exactly whenevery point of U is an interior point of U.

If there is something called an open set, surely there should be something called aclosed set and here is the definition of one.

Definition 6.2.2 A subset, C, of Fn is called a closed set if Fn \C is an open set. Theysymbol, Fn \C denotes everything in Fn which is not in C. It is also called the complementof C. The symbol, SC is a short way of writing Fn \S.

To illustrate this definition, consider the following picture.

x U

B(x,r)

You see in this picture how the edges are dotted. This is because an open set, cannot include the edges or the set would fail to be open. For example, consider what wouldhappen if you picked a point out on the edge of U in the above picture. Every open ballcentered at that point would have in it some points which are outside U . Therefore, such apoint would violate the above definition. You also see the edges of B(x,r) dotted suggesting

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