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, ξn ) by the single letter ξ .b. A Base in the Set of PartitionsIn the set P of partitions with distinguished points on a given interval [a, b], weconsider the following base B = {Bd }. The element Bd , d > 0, of the base B consistsof all partitions with distinguished points (P , ξ ) on [a, b] for which λ(P ) < d.Let us verify that {Bd }, d > 0 is actually a base in P.First Bd = ∅. In fact, for any number d > 0, it is obvious that there exists apartition P of [a, b] with mesh λ(P ) < d (for example, a partition into n congruentclosed intervals).
But then there also exists a partition (P , ξ ) with distinguishedpoints for which λ(P ) < d.Second, if d1 > 0, d2 > 0, and d = min{d1 , d2 }, it is obvious that Bd1 ∩ Bd2 =Bd ∈ B.Hence B = {Bd } is indeed a base in P.c. Riemann SumsDefinition 3 If a function f is defined on the closed interval [a, b] and (P , ξ ) is apartition with distinguished points on this closed interval, the sumσ (f ; P , ξ ) :=ni=1f (ξi )Δxi ,(6.3)3346Integrationwhere Δxi = xi − xi−1 , is the Riemann sum of the function f corresponding to thepartition (P , ξ ) with distinguished points on [a, b].Thus, when the function f is fixed, the Riemann sum σ (f ; P , ξ ) is a functionΦ(p) = σ (f ; p) on the set P of all partitions p = (P , ξ ) with distinguished pointson the closed interval [a, b].Since there is a base B in P, one can ask about the limit of the function Φ(p)over that base.d.
The Riemann IntegralLet f be a function defined on a closed interval [a, b].Definition 4 The number I is the Riemann integral of the function f on the closedinterval [a, b] if for every ε > 0 there exists δ > 0 such thatnf (ξi )Δxi < εI −i=1for any partition (P , ξ ) with distinguished points on [a, b] whose mesh λ(P ) is lessthan δ.Since the partitions p = (P , ξ ) for which λ(P ) < δ form the element Bδ of thebase B introduced above in the set P of partitions with distinguished points, Definition 4 is equivalent to the statementI = lim Φ(p),Bthat is, the integral I is the limit over B of the Riemann sums of the function fcorresponding to partitions with distinguished points on [a, b].It is natural to denote the base B by λ(P ) → 0, and then the definition of theintegral can be rewritten asI = limnλ(P )→0f (ξi )Δxi .(6.4)i=1The integral of f (x) over [a, b] is denoted bf (x) dx,ain which the numbers a and b are called respectively the lower and upper limits ofintegration.
The function f is called the integrand, f (x) dx is called the differentialform, and x is the variable of integration. Thusabf (x) dx := limλ(P )→0ni=1f (ξi )Δxi .(6.5)6.1 Definition of the Integral335Definition 5 A function f is Riemann integrable on the closed interval [a, b] if thelimit of the Riemann sums in (6.5) exists as λ(P ) → 0 (that is, the Riemann integralof f is defined).The set of Riemann-integrable functions on a closed interval [a, b] will be denoted R[a, b].Since we shall not be considering any integrals except the Riemann integral for awhile, we shall agree for the sake of brevity to say simply “integral” and “integrablefunction” instead of “Riemann integral” and “Riemann-integrable function”.6.1.3 The Set of Integrable FunctionsBy the definition of the integral (Definition 4) and its reformulation in the forms(6.4) and (6.5), an integral is the limit of a certain special function Φ(p) =σ (f ; P , ξ ), the Riemann sum, defined on the set P of partitions p = (P , ξ ) withdistinguished points on [a, b].
This limit is taken with respect to the base B in Pthat we have denoted λ(P ) → 0.Thus the integrability or nonintegrability of a function f on [a, b] depends onthe existence of this limit.By the Cauchy criterion, this limit exists if and only if for every ε > 0 there existsan element Bδ ∈ B in the base such that Φ p − Φ p < εfor any two points p , p in Bδ .In more detailed notation, what has just been said means that for any ε > 0 thereexists δ > 0 such that σ f ; P , ξ − σ f ; P , ξ < εor, what is the same, nn f ξi Δxi −f ξi Δxi < εi=1(6.6)i=1for any partitions (P , ξ ) and (P , ξ ) with distinguished points on the interval[a, b] with λ(P ) < δ and λ(P ) < δ.We shall use the Cauchy criterion just stated to find first a simple necessary condition, then a sufficient condition for Riemann integrability.a.
A Necessary Condition for IntegrabilityProposition 1 A necessary condition for a function f defined on a closed interval[a, b] to be Riemann integrable on [a, b] is that f be bounded on [a, b].336In short,6Integration f ∈ R[a, b] ⇒ f is bounded on [a, b] .Proof If f is not bounded on [a, b], then for any partition P of [a, b] the function f is unbounded on at least one of the intervals [xi−1 , xi ] of P .
This meansthat, by choosing the point ξi ∈ [xi−1 , xi ] in different ways, we can make thequantity|f (ξi )Δxi | as large as desired. But then the Riemann sum σ (f ; P , ξ ) =nf(ξi )Δxi can also be made as large as desired in absolute value by changingi=1only the point ξi in this interval.It is clear that there can be no possibility of a finite limit for the Riemann sums insuch a case. That was in any case clear from the Cauchy criterion, since relation (6.6)cannot hold in that case, even for arbitrarily fine partitions.As we shall see, the necessary condition just obtained is far from being bothnecessary and sufficient for integrability. However, it does enable us to restrict consideration to bounded functions.b.
A Sufficient Condition for Integrability and the Most Important Classesof Integrable FunctionsWe begin with some notation and remarks that will be used in the explanation tofollow.We agree that when a partition Pa = x0 < x 1 < · · · < x n = bis given on the interval [a, b], we shall use the symbol Δi to denote the interval[xi−1 , xi ] along with Δxi as a notation for the difference xi − xi−1 .If a partition P̃ of the closed interval [a, b] is obtained from the partition P bythe adjunction of new points to P , we call P̃ a refinement of P ..
of a partition P is constructed, some (perhaps all) of theWhen a refinement Pclosed intervals Δi = [xi−1 , xi ] of the partition P themselves undergo partitioning:xi−1 = xi0 < · · · < xini = xi . In that connection, it will be useful for us to labelthe points of P̃ by double indices. In the notation xij the first index means thatxij ∈ Δi , and the second index is the ordinal number of the point on the closedinterval Δi . It is now natural to set Δxij := xij − xij −1 and Δij := [xij −1 , xij ].Thus Δxi = Δxi1 + · · · + Δxini .As an example of a partition that is a refinement of both the partition P and P .
= P ∪ P , obtained as the union of the points of the two partitionsone can take PP and P .We recall finally that, as before, ω(f ; E) denotes the oscillation of the functionf on the set E, that is ω(f ; E) := sup f x − f x .x ,x ∈E6.1 Definition of the Integral337In particular, ω(f ; Δi ) is the oscillation of f on the closed interval Δi . This oscillation is necessarily finite if f is a bounded function.We now state and prove a sufficient condition for integrability.Proposition 2 A sufficient condition for a bounded function f to be integrable ona closed interval [a, b] is that for every ε > 0 there exists a number δ > 0 such thatnω(f ; Δi )Δxi < εi=1for any partition P of [a, b] with mesh λ(P ) < δ..
a refinement of P . Let us estimate theProof Let P be a partition of [a, b] and P., ξ̃ ) − σ (f ; P , ξ ). Using the notationdifference between the Riemann sums σ (f ; Pintroduced above, we can write n nni σ (f ; P̃ , ξ̃ ) − σ (f ; P , ξ ) = f (ξij )Δxij −f (ξi )Δxi =i=1 j =1i=1i=1 j =1i=1 j =1 n nnin i =f (ξij )Δxij −f (ξi )Δxij = n ni =f (ξij ) − f (ξi ) Δxij ≤i=1 j =1nin f (ξij ) − f (ξi )Δxij ≤≤i=1 j =1≤nin ω(f ; Δi )Δxij =i=1 j =1nω(f ; Δi )Δxi .i=1niIn this computation we have used the relation Δxi = j =1 Δxij and the inequality|f (ξij ) − f (ξi )| ≤ ω(f ; Δi ), which holds because ξij ∈ Δij ⊂ Δi and ξi ∈ Δi .It follows from the estimate for the difference of the Riemann sums that if thefunction satisfies the sufficient condition given in the statement of Proposition 2,then for any ε > 0 we can find δ > 0 such thatσ (f ; P̃ , ξ̃ ) − σ (f ; P , ξ ) < ε2for any partition P of [a, b] with mesh λ(P ) < δ, any refinement P̃ of P , and anychoice of the sets of distinguished points ξ and ξ̃ .Now if (P , ξ ) and (P , ξ ) are arbitrary partitions with distinguished pointson [a, b] whose meshes satisfy λ(P ) < δ and λ(P ) < δ, then, by what has just.
= P ∪ P , which is a refinement of both of them,been proved, the partition P3386Integrationmust satisfyσ (f ; P̃ , ξ̃ ) − σ f ; P , ξ < ε ,2 εσ (f ; P̃ , ξ̃ ) − σ f ; P , ξ < .2It follows that σ f ; P , ξ − σ f ; P , ξ < ε,provided λ(P ) < δ and λ(P ) < δ. Therefore, by the Cauchy criterion, the limit ofthe Riemann sums exists:nlimf (ξi )Δxi ,λ(P )→0i=1that is f ∈ R[a, b].Corollary 1 (f ∈ C[a, b]) ⇒ (f ∈ R[a, b]), that is, every continuous function ona closed interval is integrable on that closed interval.Proof If a function is continuous on a closed interval, it is bounded there, so thatthe necessary condition for integrability is satisfied in this case. But a continuousfunction on a closed interval is uniformly continuous on that interval.
Therefore, forεon any closed interval Δ ⊂every ε > 0 there exists δ > 0 such that ω(f ; Δ) < b−a[a, b] of length less than δ. Then for any partition P with mesh λ(P ) < δ we haveni=1ω(f ; Δi )Δxi <nε ε(b − a) = ε.Δxi =b−ab−ai=1By Proposition 2, we can now conclude that f ∈ R[a, b].Corollary 2 If a bounded function f on a closed interval [a, b] is continuous everywhere except at a finite set of points, then f ∈ R[a, b].Proof Let ω(f ; [a, b]) ≤ C < ∞, and suppose f has k points of discontinuity on[a, b]. We shall verify that the sufficient condition for integrability of the function fis satisfied.εFor a given ε > 0 we choose the number δ1 = 8C·kand construct the δ1 neighborhood of each of the k points of discontinuity of f on [a, b]. The complement of the union of these neighborhoods in [a, b] consists of a finite number ofclosed intervals, on each of which f is continuous and hence uniformly continuous.Since the number of these intervals is finite, given ε > 0 there exists δ2 > 0 suchthat on each interval Δ whose length is less than δ2 and which is entirely containedin one of the closed intervals just mentioned, on which f is continuous, we haveε.
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