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By W. W. Sawyer

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Prove that if a function is not continuous at x = a then it cannot be differentiable at x = a. 18. Show that {I ( lim 10 n----,,"oo nxe-nx2) dx -:/= lim n----,,"oo {I 10 (nxe- nx2 ) dx. 2 Presumptions In this book, we presume that the reader is familiar with certain notations, definitions, and theorems. The most important of these are summarized here. Notation {x E [a, b] I f (x) > O}, set notation, to the left of I is the description of the general set in which this particular set sits, to the right is the condition or conditions 16 Introduction satisfied by elements of this set.

5, show that it is Riemann integrable over [0, 1]. 7. 5, find the value of fo g(x) dx. Show the work that leads to your conclusion. 8. Prove that f is continuous at c if and only if given any E > there is a response 8 for which the oscillation of f over (c - 8, c + 8) is less than E. 9. Using the fact that a continuous function on a closed and bounded interval is uniformly continuous on that interval, prove that if f is continuous on [a, b], then f is Riemann integrable over [a, b]. 10. Find the upper and lower Darboux integrals of the characteristic function of the rationals (Example LIon page 3) over the interval [0,1].

X + h» - cos(n! 12) + h. 6. Show that 2 N-l, '"'" n. ~N!

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A Concrete Approach to Abstract Algebra by W. W. Sawyer

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