Let f(x)=0 if x is any rational number and f(x)=1 if x is any irrational number. Show that f is not integrable on 0, 1.
is not integrable on
i. If is integrable, then thefollowing limit must exist
ii. Theorem:If is continuous on , or if has only a finite number of jump discontinuities, then is integrable on
We have an interval ,
So divide it into intervals, such that
The interval always contains a rational and irrational number, however large is
Let us take as the irrational number then
Now, take as the rational number then
So here, thelimit depends on thevalue of
Thus the limit which gives integral doesn’t exist. Hence the integral doesn’t exists, that is the function is not integrable.Another proof is: Notice that this function has jump discontinuity at every rational number and irrational number. That is it has infinitely many jump discontinuities. So it is not integrable.