By David M. Bressoud

This full of life advent to degree thought and Lebesgue integration is inspired by way of the historic questions that resulted in its improvement. the writer stresses the unique goal of the definitions and theorems, highlighting the problems mathematicians encountered as those rules have been subtle. the tale starts off with Riemann's definition of the vital, after which follows the efforts of these who wrestled with the problems inherent in it, till Lebesgue ultimately broke with Riemann's definition. along with his new method of realizing integration, Lebesgue opened the door to clean and efficient techniques to the formerly intractable difficulties of research.

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**Example text**

While it may not be possible to find a Riemann sum that actually equals the upper or the lower Darboux sum, we can find Riemann sums for this partition that come arbitrarily close to the Darboux sums. The function f is Riemann integrable if and only if we can force all Riemann sums to be within E of our specified value V = f (x) dx simply by restricting our f: 2 Actually, Darboux at this time did not make a clear distinction between the supremum and maximum of a set. 26 The Riemann Integral Definition: Darboux sums Given a function f defined on [a, b] and a partition P X n = b) of this interval, we define = (a = XQ M j = sup {f(x) I Xj_1 < X < Xj} mj = inf {f (x) I x j -I < Xl < ...

15. Give an example of a function f and an interval [a, b] such that f is continuous on [a, b], differentiable at all but one point of (a, b), and for which there is no e E (a, b) for which feb) - f(a) b-a = f'ee). 16. Give an example of a function f and an interval [a, b] such that f has the intermediate value property on [a, b] but it is not continuous on this interval. 17. 5, to prove the following weaker form of Darboux's theorem: If f' is the derivative of f on an open interval containing e and if limx ---+ c - f' (x) and limx ---+ c + f' (x) exist, then these one-sided limits must be equal.

1 (Conditions for Riemann Integrability). Let f be a boundedfunction on [a, b]. This function is integrable over [a, b] if and only iffor any a > 0, a bound on the oscillation, and for any v > 0, a bound on the sum of the lengths of the intervals where the oscillation exceeds a, we can find a 8 response so that for any partition of [a, b] with subintervals of length less than 8, the subintervals on which the oscillation is at least a have a combined length that is strictly less than v. 28 The Riemann Integral The Darboux Integrals In 1881, Vito Volterra showed how to use Darboux sums to create upper and lower integrals that exist for every function.