Math 104, Section 1 Fall 2004
Sarason
HOMEWORK ASSIGNMENT 10
Due in class on Wednesday, November 24.
41. Let f:[a, b]→Rbe bounded and let g:[a, b]→Rbe Riemann integrable. Prove
U(f+g)=U(f)+U(g).
42. Let f:[a, b]→Rbe bounded and Riemann integrable over [a+, b] for every in (0,b−a).
Prove fis Riemann integrable over [a, b].
43. By a step function on [a, b]ismeant a finite linear combination of characteristic functions of
subintervals of [a, b]. Prove such a function is Riemann integrable.
44. Prove that the characteristic function of the Cantor set is Riemann integrable over [0,1] and
that its integral is 0.
45. Construct a Riemann integrable function on [0,1] that is discontinuous at each point of Q∩
(0,1). (Suggestion: To guarantee integrability, make the function nondecreasing.)