Math 421, Homework #2 Solutions
Math 421, Homework #2 Solutions (1) Let f : [a, b] R be a bounded function. Assume that f has a finite number of discontinuities, assume there exists a finite subset E of [a, b] so that f is continuous at all x [a, b] \ E. Prove that f is integrable on [a, b]. Proof. Define the set E1 = E {a, b}, and label the elements of E1 by a = q0 < q1 < < qn = b. and define a quantity q = min qj qj 1. j {1,...,n}. so that q gives the smallest distance between successive qj 's. Since f is assumed to be bounded on [a, b], there is an M > 0 so that M f (x) M for all x [a, b]. Let > 0, and define > 0 by . q = min , . 8M n 4. Define a partition Q = {x0.}
Math 421, Homework #2 Solutions (1) Let f: [a;b] !R be a bounded function. Assume that fhas a nite number of discontinuities, i.e. assume there exists a nite subset Eof [a;b] so that fis continuous at all x2[a;b] nE. Prove that fis integrable on [a;b]. Proof. De ne the set E 1 = E[fa;bg, and label the elements of E 1 by a= q 0 <q 1 < <q n= b ...
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