SEQUENCES AND SERIES OF FUNCTIONS 167 10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4, or the definition), consider the function (nx) f(x) = E (x real). Find all discontinuities of f, and show that they form a countable dense set. Show that f is nevertheless Riemann-integrable on every bounded interval. 11. Suppose {f.), {gn) are defined on E, and (a) Ef, has uniformly bounded partial sums;
SEQUENCES AND SERIES OF FUNCTIONS 167 10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4, or the definition), consider the function (nx) f(x) = E (x real). Find all discontinuities of f, and show that they form a countable dense set. Show that f is nevertheless Riemann-integrable on every bounded interval. 11. Suppose {f.), {gn) are defined on E, and (a) Ef, has uniformly bounded partial sums;
SEQUENCES AND SERIES OF FUNCTIONS 167 10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4, or the definition), consider the function (nx) f(x) = E (x real). Find all discontinuities of f, and show that they form a countable dense set. Show that f is nevertheless Riemann-integrable on every bounded interval. 11. Suppose {f.), {gn) are defined on E, and (a) Ef, has uniformly bounded partial sums;
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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