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If F is Riemann Integrable on a b and Continuous at X0 Prove F X0 is Differentiable

MATH 554 - INTEGRATION
Handout #9 - 11/20/97


Defn .   A collection of n+1 distinct points of the interval [a,b]

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is called a partition of the interval.   In this case, we define the norm of the partition by

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where tex2html_wrap_inline293 is the length of the i-th subinterval tex2html_wrap_inline297 .

Defn .   For a given partition P, we define the Riemann upper sum of a function f by

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where tex2html_wrap_inline305 denotes the supremum of f over each of the subintervals tex2html_wrap_inline297 . Similarly, we define the Riemann lower sum of a function f by

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where tex2html_wrap_inline315 denotes the infimum of f over each of the subintervals tex2html_wrap_inline297 . Since tex2html_wrap_inline321 , we note that

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for any partition P.

Defn .   Suppose tex2html_wrap_inline327 are both partitions of [a,b], then tex2html_wrap_inline331 is called a refinement of tex2html_wrap_inline333 , denoted by

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if as sets tex2html_wrap_inline337 .

Note . Iftex2html_wrap_inline339 , it follows that tex2html_wrap_inline341 since each of the subintervals formed by tex2html_wrap_inline331 is contained in a subinterval which arises from tex2html_wrap_inline333 .

Lemma .   If tex2html_wrap_inline339 , then

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and

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Proof. Suppose first that tex2html_wrap_inline333 is a partition of [a,b] and that tex2html_wrap_inline331 is the partition obtained from tex2html_wrap_inline333 by adding an additional point z. The general case follows by induction, adding one point at at time. In particular, we let

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and

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for some fixed i. We focus on the upper Riemann sum for these two partitions, noting that the inequality for the lower sums follows similarly. Observe that

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and

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where tex2html_wrap_inline373 and tex2html_wrap_inline375 . It then follows that tex2html_wrap_inline377 since

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Defn .   If tex2html_wrap_inline333 and tex2html_wrap_inline331 are arbitrary partitions of [a,b], then the common refinement of tex2html_wrap_inline333 and tex2html_wrap_inline331 is the formal union of the two.

Corollary .   Suppose tex2html_wrap_inline333 and tex2html_wrap_inline331 are arbitrary partitions of [a,b], then

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Proof. Let P be the common refinement of tex2html_wrap_inline333 and tex2html_wrap_inline331 , then

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Defn .   The lower Riemann integral of f over [a,b] is defined to be

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Similarly, the upper Riemann integral of f over [a,b] is defined to be

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By the definitions of least upper bound and greatest lower bound, it is evident that for any function f there holds

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Defn .   A function f is Riemann integrable over [a,b] if the upper and lower Riemann integrals coincide. We denote this common value by tex2html_wrap_inline427 .

Examples :

Theorem .   A necessary and sufficient condition for f to be Riemann integrable is    given tex2html_wrap_inline437 , there exists a partition P of [a,b] such that

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Proof.  First we show that (*) is a sufficient condition. This follows immediately, since for each tex2html_wrap_inline437 that there is a partition P such that (*) holds,

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Since tex2html_wrap_inline437 was arbitrary, then the upper and lower Riemann integrals of f must coincide.

To prove that (*) is a necessary condition for f to be Riemann integrable, we let tex2html_wrap_inline457 By the definition of the upper Riemann integral as a infimum of upper sums, we can find a partition tex2html_wrap_inline333 of [a,b] such that

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Similarly, we have

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Let P be a common refinement of tex2html_wrap_inline333 and tex2html_wrap_inline331 , then subtracting the two previous inequalities implies,

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Defn .   A Riemann sum for f for a partition P of an interval [a,b] is defined by

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where the tex2html_wrap_inline483 , satisfying tex2html_wrap_inline485 ), are arbitrary.

Corollary .   Suppose that f is Riemann integrable on [a,b], then there is a unique number tex2html_wrap_inline491 (tex2html_wrap_inline493 such that for every tex2html_wrap_inline437 there exists a partition P of [a,b] such that if tex2html_wrap_inline501 , then

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where tex2html_wrap_inline505 is any Riemann sum of f for the partition tex2html_wrap_inline333 . In this sense, we can interpret

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although we would actually need to show a little more to be entirely correct.

Proof.  Since tex2html_wrap_inline513 for all partitions, we see that parts i.) and ii.) follow from the definition of the Riemann integral. To see part iii.), we observe that tex2html_wrap_inline515 and hence that

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But we also know that both

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and condition (*) hold, from which part iii.) follows. tex2html_wrap_inline521

Theorem .   If f is continuous on [a,b], then f is Riemann-integrable on [a,b].

Proof.  We use the condition (*) to prove that f is Riemann-integrable. If tex2html_wrap_inline437 , we set tex2html_wrap_inline535 . Since f is continuous on [a,b], f is uniformly continuous. Hence there is a tex2html_wrap_inline543 such that tex2html_wrap_inline545 if tex2html_wrap_inline547 . Suppose that tex2html_wrap_inline549 , then it follows that tex2html_wrap_inline551 . Hence

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Theorem .   If f is monotone on [a,b], then f is Riemann-integrable on [a,b].

Proof. If f is constant, then we are done. We prove the case for f monotone increasing. The case for monotone decreasing is similiar. We again use the condition (*) to prove that f is Riemann-integrable. If tex2html_wrap_inline437 , we set tex2html_wrap_inline571 and consider any partition P with tex2html_wrap_inline549 . Since f is monotone increasing on [a,b], then tex2html_wrap_inline581 and tex2html_wrap_inline583 . Hence

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Theorem .   (Properties of the Riemann Integral) Suppose that f and g are Riemann integrable and k is a real number, then

i.) tex2html_wrap_inline593
ii.) tex2html_wrap_inline595
iii.) tex2html_wrap_inline597 implies tex2html_wrap_inline599 .
iv.) tex2html_wrap_inline601

Proof. To prove part i.), we observe that in case tex2html_wrap_inline603 , then tex2html_wrap_inline605 and tex2html_wrap_inline607 . Hence U(P,kf)=kU(P,f) and L(P,kf)=kL(P,f). In the case that k;SPMlt;0, then tex2html_wrap_inline615 and tex2html_wrap_inline617 . It follows in this case that U(P,kf)=kL(P,f) and L(P,kf)=kU(P,f) and so

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To prove property ii.) we notice that tex2html_wrap_inline627 and tex2html_wrap_inline629 for any interval I (for example, tex2html_wrap_inline633 ). Hence,

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Let tex2html_wrap_inline437 , then since f and g are Riemann integrable, there exist partitions tex2html_wrap_inline327 such that

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If we let P be a common refinement of tex2html_wrap_inline333 and tex2html_wrap_inline331 , then by combining inequalities (1) and (2), we see that see that

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Property iii.) follows directly from the definition of the upper and lower integrals using, for example, the inequality tex2html_wrap_inline655 .

Property iv.) is proved by applying property iii.) to the inequality

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from which it follows that tex2html_wrap_inline659 . But this inequality implies property iv.). tex2html_wrap_inline521

Defn .   We extend the definition of the integral to include general limits of integration which are consistent with our earlier definition.

  1. tex2html_wrap_inline663 .
  2. tex2html_wrap_inline665 .

Theorem .   If f is Riemann integrable on [a,b], then it is Riemann integrable on each subinterval tex2html_wrap_inline671 . Moreover, if tex2html_wrap_inline673 , then

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Proof. We show first that condition (*) holds for the interval [c,d]. Suppose tex2html_wrap_inline437, then by (*) applied to f over the interval [a,b], we have that there exists a partition P of [a,b] such that condition (*) holds. Let tex2html_wrap_inline689 be the refinement obtained from P which contains the points c and d. Let tex2html_wrap_inline697 be the partition obtained by restricting the partition tex2html_wrap_inline689 to the interval [c,d], then

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and so f is Riemann integrable over [c,d].

To prove the identity (3), we use the fact that condition (*) holds when f is Riemann integrable. Let tex2html_wrap_inline437 , then for tex2html_wrap_inline713 , we may apply (*) to each of the intervals I=[a,b], [a,c] and [c,b], respectively, to obtain partitions tex2html_wrap_inline719 which satisfy

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We let P be the partition of [a,b] formed by the union of the two partitions tex2html_wrap_inline727 , and tex2html_wrap_inline689 be the common refinement of P and tex2html_wrap_inline733 . Observing that

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we can combine with inequality (4) to obtain

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Since tex2html_wrap_inline437 was arbitrary, then equality (3) must hold. tex2html_wrap_inline521

Theorem .   (Intermediate Value Theorem for Integrals) If f is continuous on [a,b], then there exists tex2html_wrap_inline747 between a and b such that

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Proof.  Since f is continuous on [a,b] and for tex2html_wrap_inline759 there holds

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then by the Intermediate Value Theorem for continuous functions, there exists a tex2html_wrap_inline763 such that tex2html_wrap_inline765 . tex2html_wrap_inline521

Theorem .   (Fundamental Theorem of Calculus, Part I. Derivative of an Integral) Suppose that f is continuous on [a,b] and set tex2html_wrap_inline773 , then F is differentiable and F'(x) = f(x) for a<x<b.

Proof.  Notice that

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for some tex2html_wrap_inline747 between tex2html_wrap_inline785 and tex2html_wrap_inline787 . Hence, as tex2html_wrap_inline789 , then tex2html_wrap_inline791 converges to tex2html_wrap_inline785 and so the displayed difference quotient has a limit of tex2html_wrap_inline795 as tex2html_wrap_inline789 . tex2html_wrap_inline521

Theorem .   (Fundamental Theorem of Calculus, Part II. Integral of a Derivative) Suppose that F is function with a continuous derivative on [a,b], then

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Proof. Define tex2html_wrap_inline807 , and set H:=F-G. Since the derivative of H is identically zero by Part I of the Fundamental Theorem of Calculus, then the Mean Value Theorem implies that H(b) = H(a). Expressing this in terms of F and G gives

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which establishes the theorem. tex2html_wrap_inline521


Robert Sharpley
Fri Jan 9 23:07:34 EST 1998

robsonculdriatat.blogspot.com

Source: https://people.math.sc.edu/sharpley/math554_s96/554_9/