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. . , x m ) in Rm its ith coordinate x i . Thusπ i (x) = x i .If a = (a 1 , . . . , a m ), then obviouslyπ i (x) → a ias x → a.The function x → π i (x) does not tend to any finite value nor to infinity asx → ∞ if m > 1.On the other hand,f (x) =m i 2π (x) → ∞ as x → ∞.i=1One should not think that the limit of a function of several variables can be foundby computing successively the limits with respect to each of its coordinates. Thefollowing examples show why this is not the case.Example 2 Let the function f : R2 → R be defined at the point (x, y) ∈ R2 as follows:! xyif x 2 + y 2 = 0,22,f (x, y) = x +y0,if x 2 + y 2 = 0.4207 Functions of Several Variables: Their Limits and ContinuityThen f (0, y) = f (x, 0) = 0, while f (x, x) = 12 for x = 0.Hence this function has no limit as (x, y) → (0, 0).On the other hand,lim lim f (x, y) = lim (0) = 0,y→0 x→0y→0x→0 y→0x→0lim lim f (x, y) = lim (0) = 0.Example 3 For the functionf (x, y) =we have! x 2 −y 2x 2 +y 2, if x 2 + y 2 = 0,if x 2 + y 2 = 0,0, 2x= 1,lim lim f (x, y) = limx→0 y→0x→0 x 2 2ylim lim f (x, y) = lim − 2 = −1.y→0 x→0y→0yExample 4 For the functionf (x, y) =x + y sin x1 ,0,if x = 0,if x = 0,we havelim(x,y)→(0,0)f (x, y) = 0,lim lim f (x, y) = 0,x→0 y→0yet at the same time the iterated limitlim lim f (x, y)y→0 x→0does not exist at all.Example 5 The function!f (x, y) =x2y,x 4 +y 2if x 2 + y 2 = 0,0,if x 2 + y 2 = 0,has a limit of zero upon approach to the origin along any ray x = αt, y = βt.At the same time, the function equals 12 at any point of the form (a, a 2 ), wherea = 0, and so the function has no limit as (x, y) → (0, 0).7.2 Limits and Continuity of Functions of Several Variables4217.2.2 Continuity of a Function of Several Variables and Propertiesof Continuous FunctionsLet E be a subset of Rm and f : E → Rn a function defined on E with values in Rn .Definition 6 The function f : E → Rn is continuous at a ∈ E if for every neighborhood V (f (a)) of the value f (a) that the function assumes at a, there exists aneighborhood UE (a) of a in E whose image f (UE (a)) is contained in V (f (a)).Thusf : E → Rn is continuous at a ∈ E := = ∀V f (a) ∃UE (a) f UE (a) ⊂ V f (a) .We see that Definition 6 has the same form as Definition 1 for continuity of areal-valued function, which we are familiar with from Sect.
4.1. As was the casethere, we can give the following alternate expression for this definition:f : E → Rn is continuous at a ∈ E :== ∀ε > 0 ∃δ > 0 ∀x ∈ E d(x, a) < δ ⇒ d f (x), f (a) < ε ,or, if a is a limit point of E, f : E → Rn is continuous at a ∈ E := lim f (x) = f (a) .Ex→aAs noted in Chap. 4, the concept of continuity is of interest precisely in connection with a point a ∈ E that is a limit point of the set E on which the function f isdefined.It follows from Definition 6 and relation (7.4) that the mapping f : E → Rndefined by the relation 1fx , . . .
, x m = x −→ y = y 1 , . . . , y n = = f 1 x1, . . . , xm , . . . , f n x1, . . . , xm ,is continuous at a point if and only if each of the functions y i = f i (x 1 , . . . , x m ) iscontinuous at that point.In particular, we recall that we defined a path in Rn to be a mapping f : I → Rnof an interval I ⊂ R defined by continuous functions f 1 (x), . . . , f n (x) in the form x → y = y 1 , . . . , y n = f 1 (x), . .
. , f n (x) .Thus we can now say that a path in Rn is a continuous mapping of an intervalI ⊂ R of the real line into Rn .4227 Functions of Several Variables: Their Limits and ContinuityBy analogy with the definition of oscillation of a real-valued function at a point,we introduce the concept of oscillation at a point for a function with values in Rn .Let E be a subset of Rm , a ∈ E, and BE (a; r) = E ∩ B(a; r).Definition 7 The oscillation of the function f : E → Rr at the point a ∈ E is thequantityω(f ; a) := lim ω f ; BE (a; r) .r→+0From Definition 6 of continuity of a function, taking account of the properties ofa limit and the Cauchy criterion, we obtain a set of frequently used local propertiesof continuous functions.
We now list them.Local Properties of Continuous Functionsa) A mapping f : E → Rn of a set E ⊂ Rm is continuous at a point a ∈ E if andonly if ω(f ; a) = 0.b) A mapping f : E → Rn that is continuous at a ∈ E is bounded in some neighborhood UE (a) of that point.c) If the mapping g : Y → Rk of the set Y ⊂ Rn is continuous at a point y0 ∈ Yand the mapping f : X → Y of the set X ⊂ Rm is continuous at a point x0 ∈ X andf (x0 ) = y0 , then the mapping g ◦ f : X → Rk is defined, and it is continuous atx0 ∈ X.Real-valued functions possess, in addition, the following properties.d) If the function f : E → R is continuous at the point a ∈ E and f (a) > 0 (orf (a) < 0), there exists a neighborhood UE (a) of a in E such that f (x) > 0 (resp.f (x) < 0) for all x ∈ UE (a).e) If the functions f : E → R and g : E → R are continuous at a ∈ E, thenany linear combination of them (αf + βg) : E → R, where α, β ∈ R, their product(f · g) : E → R, and, if g(x) = 0 on E, their quotient ( fg ) : E → R are defined onE and continuous at a.Let us agree to say that the function f : E → Rn is continuous on the set E if itis continuous at each point of the set.The set of functions f : E → Rn that are continuous on E will be denotedC(E, Rn ) or simply C(E), if the range of values of the functions is unambiguouslydetermined from the context.
As a rule, this abbreviation will be used when Rn = R.πiExample 6 The functions (x 1 , . . . , x m ) −→ x i (i = 1, . . . , m), mapping Rm onto R(projections) are obviously continuous at each point a = (a 1 , . . . , a m ) ∈ Rm , sincelimx→a π i (x) = a i = π i (a).Example 7 Any function x → f (x) defined on R, for example x → sin x, can alsoFbe regarded as a function (x, y) −→ f (x) defined, say, on R2 . In that case, if f was7.2 Limits and Continuity of Functions of Several Variables423Fcontinuous as a function on R, then the new function (x, y) −→ f (x) will be continuous as a function on R2 . This can be verified either directly from the definitionof continuity or by remarking that the function F is the composition (f ◦ π 1 )(x, y)of continuous functions.In particular, it follows from this, when we take account of c) and e), that thefunctions f (x, y) = sin x + exy ,f (x, y) = arctan ln |x| + |y| + 1 ,for example, are continuous on R2 .We remark that the reasoning just used is essentially local, and the fact that thefunctions f and F studied in Example 7 were defined on the entire real line R orthe plane R2 respectively was purely an accidental circumstance.Example 8 The function f (x, y) of Example 2 is continuous at any point of thespace R2 except (0, 0).
We remark that, despite the discontinuity of f (x, y) at thispoint, the function is continuous in either of its two variables for each fixed value ofthe other variable.Example 9 If a function f : E → Rn is continuous on the set E and Ẽ is a subsetof E, then the restriction f |Ẽ of f to this subset is continuous on Ẽ, as followsimmediately from the definition of continuity of a function at a point.We now turn to the global properties of continuous functions. To state them forfunctions f : E → Rn , we first give two definitions.Definition 8 A mapping f : E → Rn of a set E ⊂ Rm into Rn is uniformly continuous on E if for every ε > 0 there is a number δ > 0 such that d(f (x1 ), f (x2 )) < εfor any points x1 , x2 ∈ E such that d(x1 , x2 ) < δ.As before, the distances d(x1 , x2 ) and d(f (x1 ), f (x2 )) are assumed to be measured in Rm and Rn respectively.When m = n = 1, this definition is the definition of uniform continuity ofnumerical-valued functions that we already know.Definition 9 A set E ⊂ Rm is pathwise connected if for any pair of its points x0 ,x1 , there exists a path Γ : I → E with support in E and endpoints at these points.In other words, it is possible to go from any point x0 ∈ E to any other pointx1 ∈ E without leaving E.Since we shall not be considering any other concept of connectedness for a setexcept pathwise connectedness for the time being, for the sake of brevity we shalltemporarily call pathwise connected sets simply connected.Definition 10 A domain in Rm is an open connected set.4247 Functions of Several Variables: Their Limits and ContinuityExample 10 An open ball B(a; r), r > 0, in Rm is a domain.
We already know thatB(a; r) is open in Rm . Let us verify that the ball is connected. Let x0 = (x01 . . . , x0m )and x1 = (x11 , . . . , x1m ) be two points of the ball. The path defined by the functionsx i (t) = tx1i + (1 − t)x0i (i = 1, . . . , m), defined on the closed interval 0 ≤ t ≤ 1, hasx0 and x1 as its endpoints. In addition, its support lies in the ball B(a; r), since, byMinkowski’s inequality, for any t ∈ [0, 1],445 m5 m 52 5 i2ii6d x(t), a =x (t) − a = 6t x1 − a i + (1 − t) x0i − a i≤i=1i=1i=1i=1445 m5 m5 5 22t x1i − a i(1 − t) x0i − a i≤6+6=445 m5 m5 5 22=t ·6x1i − a i + (1 − t) · 6x0i − a i < tr + (1 − t)r = r.i=1i=1Example 11 The circle (one-dimensional sphere) of radius r > 0 is the subset of R2given by the equation (x 1 )2 + (x 2 )2 = r 2 .
Setting x 1 = r cos t, x 2 = r sin t, we seethat any two points of the circle can be joined by a path that goes along the circle.Hence a circle is a connected set. However, this set is not a domain in R2 , since it isnot open in R2 .We now state the basic facts about continuous functions in the large.Global Properties of Continuous Functionsa) If a mapping f : K → Rn is continuous on a compact set K ⊂ Rm , then it isuniformly continuous on K.b) If a mapping f : K → Rn is continuous on a compact set K ⊂ Rm , then it isbounded on K.c) If a function f : K → R is continuous on a compact set K ⊂ Rm , then itassumes its maximal and minimal values at some points of K.d) If a function f : E → R is continuous on a connected set E and assumes thevalues f (a) = A and f (b) = B at points a, b ∈ E, then for any C between A andB, there is a point c ∈ E at which f (c) = C.Earlier (Sect.
4.2), when we were studying the local and global properties offunctions of one variable, we gave proofs of these properties that extend to the moregeneral case considered here. The only change that must be made in the earlierproofs is that expressions of the type |x1 − x2 | or |f (x1 ) − f (x2 )| must be replacedby d(x1 , x2 ) and d(f (x1 ), f (x2 )), where d is the metric in the space where thepoints in question are located. This remark applies fully to everything except thelast statement d), whose proof we now give.7.2 Limits and Continuity of Functions of Several Variables425Proof d) Let Γ : I → E be a path that is a continuous mapping of an interval[α, β] = I ⊂ R such that Γ (α) = a, Γ (β) = b.
By the connectedness of E thereexists such a path. The function f ◦ Γ : I → R, being the composition of continuous functions, is continuous; therefore there is a point γ ∈ [α, β] on the closedinterval [α, β] at which f ◦ Γ (γ ) = C. Set c = Γ (γ ). Then c ∈ E and f (c) = C. Example 12 The sphere S(0; r) defined in Rm by the equation 1 2 2x+ · · · + xm = r 2,is a compact set.Indeed, it follows from the continuity of the function 1 2 2x , .
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