Darboux and Integrability Reformulation.pdf

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Unformatted text preview: CHAPTER (a, 0) FIGURE 1 FIGURE 2 INTEGRALS The derivative does not display its full strength until allied with the “integral,” the second main concept of Part III. At first this topic may seem to be a com- plete digression—in this chapter derivatives do not appear even once! The study of integrals does require a long preparation, but once this preliminary work has been completed, integrals will be an invaluable tool for creating new functions, and the derivative will reappear in Chapter 14, more powerful than ever. Although ultimately to be defined in a quite complicated way, the integral formalizes a simple, intuitive concept—that of area. By now it should come as no surprise to learn that the definition of an intuitive concept can present great difficulties—“area” is certainly no exception. In elementary geometry, formulas are derived for the areas of many plane figures, but a little reflection shows that an acceptable definition of area is seldom given. The area of a region is sometimes defined as the number of squares, with sides of length l, which fit in the region. But this definition is hopelessly inadequate for any but the simplest regions. For example, a circle of radius 1 supposedly has as area the irrational number 71', but it is not at all clear what “71' squares” means. Even if we consider a circle of radius 1/\/7r, which supposedly has area 1, it is hard to say in what way a unit square fits in this circle, since it does not seem possible to divide the unit square into pieces which can be arranged to form a circle. In this chapter we will only try to define the area of some very special regions (Figure 1)~those which are bounded by the horizontal axis, the verti- cal lines through (a, 0) and (b, 0), and the graph of a function f such that f(x) 2 0 for all x in [0, b]. It is convenient to indicate this region by R(f, a, b). Notice that these regions include rectangles and triangles, as well as many other important geometric figures. The number which we will eventually assign as the area ofR(f, a, b) will be called the integral of f on [a, 1)]. Actually, the integral will be defined even for functions f which do not satisfy the condition f(x) 2 0 for all x in [1.1, b]. Iff is the function graphed in Figure 2, the integral will represent the difference of the area of the lightly shaded region and the area of the heavily shaded region (the “algebraic area” of R(f, a, 17)). The idea behind the prospective definition is indicated in Figure 3. The interval [51, b] has been divided into four subintervals U0: til [tn t2] [[2, is] [13, t4] by means of numbers to, t1, t2, 33, £4 with a=to<l1<t2<l3<t4=b 214 FIGURE 3 DEFINITION DEFINITION Integrals 215 (the numbering of the subscripts begins with 0 so that the largest subscript will equal the number of subintervals). On the first interval [150, t1] the function f has the minimum value m1 and the maximum value M1; similarly, on the ith interval [t,-_1, ti] let the minimum value off be m,» and let the maximum value be Mi. The sum 3 = m‘(t1 — to) + 7712(62 "‘ t1) + "1303 — 52) + ”1404 —‘ 53) represents the total area of rectangles lying inside the region R(f, a, 1)), while the sum S = M1051 ‘ to) ‘l‘ M202 — £1) + M303 — t2) + M404 — l3) represents the total area of rectangles containing the region R(f, a, b). The guiding principle of our attempt to define the area A of R( f, a, b) is the obser- vation that A should satisfy 53A and ASS, and that this should be true, no matter how the interval [(1, b] is subdivided. It is to be hoped that these requirements will determine A. The following definitions begin to formalize, and eliminate some of the implicit assumptions in, this discussion. Let a < b. A partition of the interval [0, b] is a finite collection of points in [(2, b], one of which is a, and one of which is b. L. The points in a partition can be numbered to, . . . , 25,, so that a=lo<t1< ‘ ' ' <tn_1<tn=b; we shall always assume that such a numbering has been assigned. Suppose f is bounded on [(1, b] and P = {to, . . . , tn} is a partition of [51, 1)]. Let m, = inf {f(x): ti_1 S x 5 1,}, M,- = sup {f(x): ti—l S x S 5‘}- The lower sum of f for P, denoted by L( f, P), is defined as LMH=2mm—my The upper sum off for P, denoted by U (f, P), is defined as UMH=2MM—md The lower and upper sums correspond to the sums s and S in the previous 216 Derivatives and Integrals example; they are supposed to represent the total areas of rectangles lying below and above the graph off. Notice, however, that despite the geometric motivation, these sums have been defined precisely Without any appeal to a concept of “area.” Two details of the definition deserve comment. The requirement that f be bounded on [(2, b] is essential in order that all the m and M.- be defined. Note, also, that it was necessary to define the numbers m.- and M.- as inf’s and sup’s, rather than as minima and maxima, since f was not assumed continuous. One thing is clear about lower and upper sums: If P is any partition, then L(f, P) S U(f, P), because 71. LMH=2mm—m% i=1 Umm=jmefim, and for each i we have mi<£i " kg) S MM; — l‘i—1)- FIGURE 6 Supfiose first that f(x) = c foi' all x in [a, b] (Figure 6). If P = {A}, . . is any partition of [0, b], then m,- = M; = C, so 11 ML P) = Z w.- — m) = c<b — a), i=1 U0: P) = Z c<z.- — :H) = ca: — a). '=1 In this case, all lower sums and upper sums are equal, and sup {L(f: P)} = inf {U(f, Pl} = 6(1) - (I)- Now consider (Figure 7) the function f defined by O x irrational f(x) = { ’ . 1, x ratlonal. .,¢,.} Integrals 219 If P = {10, . . . , in} is any partition, then m,- = 0, since there is an irrational number in [n-1, ti], and M, = 1, since there is a rational number in [t,~_1, ti]. Therefore, 71. ufin=20wwwm=m UMB=Elww+J=b—m =1 FIGURE 7 Thus, in this case it is certainly not true that sup {L(f, P)} = inf {U(f, P) }. The principle upon which the definition of area was to be based provides insufficient information to determine a specific area for R( f, a, b)—any num- ber between 0 and b —- a seems equally good. On the other hand, the region R(f, a, b) is so weird that we might with justice refuse to assign it any area at all. In fact, we can maintain, more generally, that whenever SUP lL(f, P)} ¢ inf {U(f, 13)}, the region R( f, a, b) is too unreasonable to deserve having an area. As our appeal to the word “unreasonable” suggests, we are about to cloak our ignorance in terminology. DEFINITION A function f which is bounded on [[1, b] is integrable on [(1, b] if sup {L(f, P): P a partition of [0, b]} = inf {U(f, P): P a partition of [(1, b]}. In this case, this common number is called the integral of f on [(1, b] and is denoted by b L f- (The symbol f is called an integral sign and was originally an elongated s, for “sum;” the numbers a and b are called the lower and upper limits of integration.) The integral [abf is also called the area of R(f, a, b) when f(x) 2 0 for all x in [52, b]. If f is integrable, then according to this definition, L( f, P) g [0” f g U(f, P) for all partitions P of [(1, b]. b . . . . Moreover, f f 15 the unique number With this property. This definition merely pinpoints, and does not solve, the problem discussed before: we do not know which functions are integrable (nor do we know how to find the integral off on [a, b] when f is integrable). At present we know only 220 Derivatives and Integrals THEOREM 2 PROOF two examples: (1) iff(x) = c, then fis integrable on [(1, b] and [ff = c - (b -- a). (Notice that this integral assigns the expected area to a rectangle.) (2) iff(x) = 0, xirrational 1, x rational, then f is not integrable on [a, b]. Several more examples will be given before discussing these problems fur- ther. Even for these examples, however, it helps to have the following simple criterion for integrability stated explicitly. Iff is bounded on [a, b], then f is integrable on [(1, 1)] if and only if for every 5 > 0 there is a partition P of [(2, (2] such that U<faP)_L(f3P)<E Suppose first that for every 5 > 0 there is a partition P with Ulf, P) — L(f, P) < 6- Since inf {U(f, P’)} 3 ML P), SUP {1405199} 2 Llf, P), it follows that inf {U02 P’)} - SUP {LUZ P’)} < 8- Since this is true for all e > 0, it follows that SUP {L(f, P’)} = inf {U(f, P')}; by definition, then, f is integrable. The proof of the converse assertion is simi- lar. If f is integrable, then SUP {Llfl PM = inf {Ulfi P)}- This means that for each 8 > 0 there are partitions P’, P" with (17(f3 PH) _ L(fs PI) < 5. Let P be a partition which contains both P' and P”. Then, according to the lemma, Ulf, P) S U(f, P”), LU, P) 2 UL P'); consequently, U(f9-P) _L(f,P)S U(f:P,l)—L(f3P,) <E'I Although the mechanics of the proof take up a little space, it should be clear that Theorem 2 amounts to nothing more than a restatement of the definition Integrals 221 of integrability. Nevertheless, it is a very convenient restatement because there is no mention of sup’s and inf’s, which are often difficult to work with. The ...
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