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. .). They form an open covering of Rm and consequently alsoof K. If K were not bounded, it would be impossible to select a finite covering ofK from this sequence.Proposition 5 The set K ⊂ Rm is compact if and only if K is closed and boundedin Rm .Proof The necessity of these conditions was proved in Propositions 3 and 4.Let us verify that the conditions are sufficient. Since K is a bounded set, thereexists an m-dimensional interval I containing K.
As was shown in Example 13,I is compact in Rm . But if K is a closed set contained in the compact set I , then byProposition 3b) it is itself compact.7.1.4 Problems and Exercises1. The distance d(E1 , E2 ) between the sets E1 , E2 ⊂ Rm is the quantityd(E1 , E2 ) :=infx1 ∈E1 ,x2 ∈E2d(x1 , x2 ).Give an example of closed sets E1 and E2 in Rm having no points in common forwhich d(E1 , E2 ) = 0.2. Show thata) the closure E in Rm of any set E ⊂ Rm is a closed set in Rm ;b) the set ∂E of boundary points of any set E ⊂ Rm is a closed set;c) if G is an open set in Rm and F is closed in Rm , then G\F is open in Rm .3.
Show that if K1 ⊃ K2 ⊃ · · · ⊃ Kn ⊃ · · · is a sequence of nested nonempty compact sets, then ∞i=1 Ki = ∅.4. a) In the space Rk a two-dimensional sphere S 2 and a circle S 1 are situated sothat the distance from any point of the sphere to any point of the circle is the same.Is this possible?b) Consider problem a) for spheres S m , S n of arbitrary dimension in Rk . Underwhat relation on m, n, and k is this situation possible?4167 Functions of Several Variables: Their Limits and Continuity7.2 Limits and Continuity of Functions of Several Variables7.2.1 The Limit of a FunctionIn Chap. 3 we studied in detail the operation of passing to the limit for a real-valuedfunction f : X → R defined on a set in which a base B was fixed.In the next few sections we shall be considering functions f : X → Rn definedon subsets of Rm with values in R = R1 or more generally in Rn , n ∈ N.
We shallnow make a number of additions to the theory of limits connected with the specificsof this class of functions.However, we begin with the basic general definition.Definition 1 A point A ∈ Rn is the limit of the mapping f : X → Rn over a base Bin X if for every neighborhood V (A) of the point there exists an element B ∈ B ofthe base whose image f (B) is contained in V (A).In brief, lim f (x) = A := ∀V (A) ∃B ∈ B f (B) ⊂ V (A) .BWe see that the definition of the limit of a function f : X → Rn is exactly thesame as the definition of the limit of a function f : X → R if we keep in mind whata neighborhood V (A) of a point A ∈ Rn is for every n ∈ N.Definition 2 A mapping f : X → Rn is bounded if the set f (X) ⊂ Rn is boundedin Rn .Definition 3 Let B be a base in X.
A mapping f : X → Rn is ultimately boundedover the base B if there exists an element B of B on which f is bounded.Taking these definitions into account and using the same reasoning that we gavein Chap. 3, one can verify without difficulty thata function f : X → Rn can have at most one limit over a given base B in X;a function f : X → Rn having a limit over a base B is ultimately bounded overthat base.Definition 1 can be rewritten in another form making explicit use of the metricin Rn , namelyDefinition 1 lim f (x) = A ∈ Rn := ∀ε > 0 ∃B ∈ B ∀x ∈ B d f (x), A < εBor7.2 Limits and Continuity of Functions of Several Variables417Definition 1 lim f (x) = A ∈ Rn := lim d f (x), A = 0 .BBThe specific property of a mapping f : X → Rn is that, since a point y ∈ Rn isan ordered n-tuple (y 1 , . .
. , y n ) of real numbers, defining a function f : X → Rnis equivalent to defining n real-valued functions f i : X → R (i = 1, . . . , n), wheref i (x) = y i (i = 1, . . . , n).If A = (A1 , . . . , An ) and y = (y 1 , . . . , y n ), we have the inequalities i√y − Ai ≤ d(y, A) ≤ n max y i − Ai ,(7.3)1≤i≤nfrom which one can see thatlim f (x) = A ⇔ lim f i (x) = AiBB(i = 1, . . . , n),(7.4)that is, convergence in Rn is coordinatewise.Now let X = N be the set of natural numbers and B the base k → ∞, k ∈ N,in X. A function f : N → Rn in this case is a sequence {yk }, k ∈ N, of points of Rn .Definition 4 A sequence {yk }, k ∈ N, of points yk ∈ Rn is fundamental (a Cauchysequence) if for every ε > 0 there exists a number N ∈ N such that d(yk1 , yk2 ) < εfor all k1 , k2 > N .One can conclude from the inequalities (7.3) that a sequence of points yk =1(yk , .
. . , ykn ) ∈ Rn is a Cauchy sequence if and only if each sequence of coordinateshaving the same labels {yki }, k ∈ N, i = 1, . . . , n, is a Cauchy sequence.Taking account of relation (7.4) and the Cauchy criterion for numerical sequences, one can now assert that a sequence of points Rn converges if and onlyif it is a Cauchy sequence.In other words, the Cauchy criterion is also valid in Rn .Later on we shall call metric spaces in which every Cauchy sequence has a limitcomplete metric spaces.
Thus we have now established that Rn is a complete metricspace for every n ∈ N.Definition 5 The oscillation of a function f : X → Rn on a set E ⊂ X is the quantityω(f ; E) := d f (E) ,where d(f (E)) is the diameter of f (E).As one can see, this is a direct generalization of the definition of the oscillationof a real-valued function, which Definition 5 becomes when n = 1.The validity of the following Cauchy criterion for the existence of a limit forfunctions f : X → Rn with values in Rn results from the completeness of Rn .4187 Functions of Several Variables: Their Limits and ContinuityTheorem 1 Let X be a set and B a base in X.
A function f : X → Rn has a limitover the base B if and only if for every ε > 0 there exists an element B ∈ B of thebase on which the oscillation of the function is less than ε.Thus,∃ lim f (x) ⇔ ∀ε > 0 ∃B ∈ B ω(f ; B) < ε .BThe proof of Theorem 1 is a verbatim repetition of the proof of the Cauchy criterion for numerical functions (Theorem 4 in Sect. 3.2), except for one minor change:|f (x1 ) − f (x2 )| must be replaced throughout by d(f (x1 ), f (x2 )).One can also verify Theorem 1 another way, regarding the Cauchy criterion asknown for real-valued functions and using relations (7.4) and (7.3).The important theorem on the limit of a composite function also remains validfor functions with values in Rn .Theorem 2 Let Y be a set, BY a base in Y , and g : Y → Rn a mapping having alimit over the base BY .Let X be a set, BX a base in X, and f : X → Y a mapping of X into Y such thatfor each BY ∈ BY there exists BX ∈ BX such that the image f (BX ) is containedin BY .Under these conditions the composition g ◦ f : X → Rn of the mappings f andg is defined and has a limit over the base BX , andlim(g ◦ f )(x) = lim g(y).BXBYThe proof of Theorem 2 can be carried out either by repeating the proof of Theorem 5 of Sect.
3.2, replacing R by Rn , or by invoking that theorem and using relation(7.4).Up to now we have considered functions f : X → Rn with values in Rn , withoutspecifying their domains of definition X in any way. From now on we shall primarilybe interested in the case when X is a subset of Rm .As before, we make the following conventions.U (a) is a neighborhood of the point a ∈ Rm ;Ů (a) is a deleted neighborhood of a ∈ Rm , that is, Ů (a) := U (a)\a;UE (a) is a neighborhood of a in the set E ⊂ Rm , that is, UE (a) := E ∩ U (a);ŮE (a) is a deleted neighborhood of a in E, that is, ŮE (a) := E ∩ Ů (a);x → a is the base of deleted neighborhoods of a in Rm ;x → ∞ is the base of neighborhoods of infinity, that is, the base consisting of thesets Rm \B(a; r);x → a, x ∈ E or (E x → a) is the base of deleted neighborhoods of a in E ifa is a limit point of E;x → ∞, x ∈ E or (E x → ∞) is the base of neighborhoods of infinity in Econsisting of the sets E\B(a; r), if E is an unbounded set.7.2 Limits and Continuity of Functions of Several Variables419In accordance with these definitions, one can, for example, give the followingspecific form of Definition 1 for the limit of a function when speaking of a functionf : E → Rn mapping a set E ⊂ Rm into Rn : lim f (x) = A := ∀ε > 0 ∃ŮE (a) ∀x ∈ ŮE (a) d f (x), A < ε .Ex→aThe same thing can be written another way:lim f (x) = A :=Ex→a= ∀ε > 0 ∃δ > 0 ∀x ∈ E 0 < d(x, a) < δ ⇒ d f (x), A < ε .Here it is understood that the distances d(x, a) and d(f (x), A) are measured in thespaces (Rm and Rn ) in which these points lie.Finally, lim f (x) = A := ∀ε > 0 ∃B(a; r) ∀x ∈ Rm \B(a; r) d f (x), A < ε .x→∞Let us also agree to say that, in the case of a mapping f : X → Rn , the phrase“f (x) → ∞ in the base B” means that for any ball B(A; r) ⊂ Rn there exists B ∈ Bof the base B such that f (B) ⊂ Rn \B(A; r).Example 1 Let x → π i (x) be the mapping π i : Rm → R assigning to each x =(x 1 , .
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