[PDF] [PDF] Darboux sums and Riemann-integrability Definition of Darboux

[PDF] Darboux sums and Riemann-integrability Definition of Darboux

Darboux sums and Riemann-integrability Definition of Darboux sums Let f be a bounded function defined on a closed bounded interval [a, b] A partition P of [a, 

[PDF] Darboux Sums - UMD MATH

Math 410 Section 6 1: Darboux Sums - Lower and Upper Integrals 1 (f) Definition: If P is a partition of [a, b] then a refinement of P is a partition containing at

Monotonicity Properties of Darboux Sums

f(ξj)(xj − xj−1) Riemann's definition of the definite integral says that f is integrable in the interval [a, b], if the sums S(f 

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Definition of lower and upper Darboux sums Let f : [a, b] → R (with a < b) be a bounded function For any partition P = ((t0, ,tn)) of [a, b] the • lower Darboux sum 

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Darboux sums and partitions Let f : [a, b] → R be a bounded function For a partition P Definition of integral We define the upper and lower integrals by ∫ ¯b a

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21 jan 2020 · definition of “an integral” in terms of Cauchy sums (which are left-Riemann We define the upper Darboux sum of with respect to by

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our definition of the Darboux integral will not depend upon this assumption possible ways to partition the interval [a, b], the supremum of the lower sums so 

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We will introduce Riemann sums and Darboux sums, then two definitions of integrability Then the Darbows upper sum is defined by Ulf P) = ² Mest and

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The basic idea underlying the definition is to compute the area under the graph of partition P as above, the upper Darboux sum is defined by U(f, P) := n В

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Define M(f,S) = sup{f(x) x ∈ S} and m(f,S) = inf{f(x) x ∈ S} Given a partition P = {a = t0 < t1 < t2

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Real Analysis

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Darboux sums and Riemann-integrability

Denition of Darboux sums.

Letfbe a bounded function dened on a closed bounded interval[a; b]: A partitionPof[a; b]is any nite selection of pointsa=x0We say thatP′is a renement ofPifP?P′: LetmkandMkbe the inmum and the supremum offover[xk-1; xk]; k=infx?[xk-1;xk]f(x) k=sup x?[xk-1;xk]f(x): Consider the lower and upper Darboux sums offcorresponding to a partitionP?

L(f;P)=n

k=1m k(xk-xk-1)

U(f;P)=n

k=1M k(xk-xk-1):

Properties of Darboux sums.

?Lower and upper bounds ?Monotonicity with respect to the partition ?Inequality of lower and upper sums

Lower and upper Darboux integrals.

The lower integral offover[a; b]is the quantityL(f)=supPL(f;P): The upper integral offover[a; b]is the quantityU(f)=infPU(f;P):

Riemann integrability.

One says thatfis integrable in the sense of Riemann ifL(f)=U(f):

The common value is denoted by

af(x)dxor just by?b af:

Exercise. Consider the indicator functionf(x)=1?1

?(x)on[0;1]:

DetermineL(f;P); U(f;P); L(f); U(f):

Exercise. For a nite setE={a1;:::;am}?[0;1];consider the indicator function f(x)=1E(x)on[0;1]:DetermineL(f;P); U(f;P); L(f); U(f): Exercise. Consider the Dirichlet function, the indicator function of rational points f(x)=1Q(x)on[0;1]:DetermineL(f;P); U(f;P); L(f); U(f): The preceding exercise shows that the Dirichlet function is not Riemann-integrable. So a pointwise limit of Riemann-integrable functions is not necessarily Riemann integrable.

More properties of Darboux sums.

?Monotonicity with respect to the function As a consequence, this property holds for the lower and upper integral. ?Positive homogeneity Forc>0; L(cf;P)=cL(f;P)andU(cf;P)=cU(f;P):Note:L(-f;P)=-U(f;P): ?Uniform limits Iffn⇉fon[a; b];thenL(fn;P)→L(f;P)andU(fn;P)→U(f;P): Proof. Given">0;choose an indexNso that?fn-f?<"forn>N: So the sequenceL(fn;P)converges toL(f;P)asn→∞: The case of the upper sum is treated analogously.◻ ?Superadditivity and subadditivity with respect to the function As a consequence, the lower integral is superadditive and the upper integral is subadditive.

The Riemann integral is therefore additive.

Criterion for Riemann integrability.

A functionfis Riemann integrable on[a; b]if and only if for every">0 there is a partitionPof[a; b]such thatU(f;P)-L(f;P)<":

Sequential criterion for Riemann integrability.

A functionfis Riemann integrable on[a; b]if and only if there is a sequence of partitionsPnof[a; b]such that limn→∞L(f;P)=limn→∞U(f;P):

The class of Riemann-integrable functions.

The class includes continuous functions, functions with nitely many jump discontinuities (step functions, in particular), monotone functions. To give a full description of the class, we would need a couple of new instruments.quotesdbs_dbs2.pdfusesText_4