Answer to Question #91394 in Calculus for Sajid

Question

Q. choose the correct option.
Q. Which of the following statement is not true?
a. If f is bounded on [a,b], then f is Reimann integrable on [a,b].
b. If f is monotone on [a,b], then f is Reimann integrable on [a,b].
c. If f is continuous on [a,b], then f is Reimann integrable on [a,b].
d. If f is Lebesgue measurable on [a,b], then f is Reimann integrable on [a,b].

Answer 1

Statement a) is false. Confirm two examples from the previous answer (Dirichlet function and 
f (x) = 1) Both functions are bounded. The Dirichlet function is discontinuous at all points of the segment [0,1]. The Lebesgue measure of the set of discontinuity points is not equal to zero. Consequently, the Dirichlet function is not integrable. The function f (x) = 1 is integrable, since it is continuous.
Another variant of the proof of non-integrability of the Dirichlet function: all upper integral sums are equal to 1, lower ones are equal to 0. So the upper and lower Riemann integrals do not coincide.
Statement d) is incorrect. A counterexample is the Lebesgue measurable Dirichlet function. (characteristic function of a Lebesgue measurable set of rational numbers)
Statement b) is true: A continuous function on a segment is bounded and has no discontinuity points. Therefore, by Lebesgue criterion, a continuous function is Riemann integrable 
Statement c) is true: the monotone function on the segment   is bounded, the set of discontinuity points is at most countable, therefore the Lebesgue measure of this set is 0.