12.8. OPEN AND CLOSED SETS 229

17. Let f (x,y,z) = x2y+ sin(xyz). Does f achieve a maximum on the set{(x,y,z) : x2 + y2 +2z2 ≤ 8

}?

Explain why.

18. Suppose x is defined to be a limit point of a set A if and only if for all r > 0, B(x,r)contains a point of A different than x. Show this is equivalent to the above definitionof limit point.

19. Give an example of an infinite set of points in R3 which has no limit points. Showthat if D(f) equals this set, then f is continuous. Show that more generally, if f isany function for which D(f) has no limit points, then f is continuous.

20. Let {xk}nk=1 be any finite set of points in Rp. Show this set has no limit points.

21. Suppose S is any set of points such that every pair of points is at least as far apart as1. Show S has no limit points.

22. Find limx→0sin(|x|)|x| and prove your answer from the definition of limit.

23. Suppose g is a continuous vector valued function of one variable defined on [0,∞).Prove

limx→x0

g (|x|) = g (|x0|) .

12.8 Open And Closed SetsEventually, one must consider functions which are defined on subsets of Rn and their prop-erties. The next definition will end up being quite important. It describe a type of subset ofRn with the property that if x is in this set, then so is y whenever y is close enough to x.

Definition 12.8.1 Recall that for x,y ∈ Rn,

|x−y|=

(n

∑i=1|xi− yi|2

)1/2

.

Also letB(x,r)≡ {y ∈ Rn : |x−y|< r}

Let U ⊆ Rn. U is an open set if whenever x ∈U, there exists r > 0 such that B(x,r)⊆U.More generally, if U is any subset of Rn, x∈U is an interior point of U if there exists r > 0such that x ∈ B(x,r)⊆U. In other words U is an open set exactly when every point of Uis 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 12.8.2 A subset, C, of Rn is called a closed set if Rn \C is an open set. Theysymbol Rn \C denotes everything in Rn which is not in C. It is also called the complementof C. The symbol SC is a short way of writing Rn \S.