Chapter 16

Functions Of Many Variables

16.1 Review Of LimitsRecall the concept of limit of a function of many variables. When f : D(f)→ Rq one canonly consider in a meaningful way limits at limit points of the set D(f).

Definition 16.1.1 Let A denote a nonempty subset of Rp. A point x is said to be a limitpoint of the set A if for every r > 0,B(x,r) contains infinitely many points of A.

Example 16.1.2 Let S denote the set{(x,y,z) ∈ R3 : x,y,z are all in N

}. Which points are

limit points?

This set does not have any because any two of these points are at least as far apart as 1.Therefore, if x is any point of R3,B(x,1/4) contains at most one point.

Example 16.1.3 Let U be an open set in R3. Which points of U are limit points of U?

They all are. From the definition of U being open, if x ∈U , There exists B(x,r)⊆Ufor some r > 0. Now consider the line segment x+ tre1 where t ∈ [0,1/2]. This describesinfinitely many points and they are all in B(x,r) because |x+ tre1−x|= tr< r. Therefore,every point of U is a limit point of U .

The case where U is open will be the one of most interest, but many other sets havelimit points.

Definition 16.1.4 Let f : D(f) ⊆ Rp → Rq where q, p ≥ 1 be a function and let x be alimit point of D(f). Then

limy→x

f (y) =L

if and only if the following condition holds. For all ε > 0 there exists δ > 0 such that if

0 < |y−x|< δ and y ∈ D(f)

then,|L−f (y)|< ε.

The condition that x must be a limit point of D(f) if you are to take a limit at x is whatmakes the limit well defined.

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