By David M. Bressoud

This vigorous advent to degree conception and Lebesgue integration is inspired by means of the old questions that ended in its improvement. the writer stresses the unique goal of the definitions and theorems, highlighting the problems mathematicians encountered as those principles have been sophisticated. the tale starts off with Riemann's definition of the imperative, after which follows the efforts of these who wrestled with the problems inherent in it, until eventually Lebesgue eventually broke with Riemann's definition. along with his new approach of figuring out integration, Lebesgue opened the door to clean and effective techniques to the formerly intractable difficulties of study.

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**Example text**

3 , Quite a bit more work. Darboux's original justification, published in 1875, had several flaws. He published an addendum in 1879 in which he corrected the justification of his original example and gave a simpler example, L~I cos(n! x)/n! 9). 2 Nondifferentiable Integrals That same year of 1875, Paul du Bois-Reymond published Weierstrass's example of an everywhere continuous but nowhere differentiable function. 3. 4 Weierstrass had publicly presented this example to the Berlin Academy in 1872, but it had not appeared in print.

9. Show that the set of all real-valued continuous functions defined on [0, 1] is a vector space. 10. Use the nested interval principle to prove that every Cauchy sequence converges. 11. Show that the nested interval principle does not necessarily hold if we replace closed intervals with open intervals. 12. 3). Show that x E (y Sk r =} x E (} sf and x ¢ ( Yr Sk =} x ¢ (} sf. 13. 4). 14. Prove that given any two sets F I and F2 , if SI = FI n FF and S2 = Fie n F2 , then F I = (F2 U SI) nSf. 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).

The variable h depends on N. 5. Show that for every positive integer n, cos(n! (x + h» - cos(n! 12) + h. 6. Show that 2 N-l, '"'" n. ~N!