Answer to Question #199236 in Calculus for Sarita bartwal

Question

The function,f: [-1,1]×[2,1] → R, defined by

f(x,y)= { x ; y is rational

{ 0 ; y is irrational

is integrable or not.

True or false with full explanation

Answer 1

A Riemann integrable bounded function on a compact interval [a, b] is continuous practically everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).

The function in our situation is discontinuous at every irrational point. The function is not integrable because the collection of irrational points on a segment has the cardinality of the continuum.

The answer is that function is not integrable.