A non-integrable function

Subject: Calculus Proof

Mark Spilker
college
student

I have a proof that I cannot do here it goes.
Let F(x)= 1 if x is a rational number 0 if x is an irrational number Prove the function is not intregrable on the interval (0,1). Hint: Show that no matter how small the norm of the partition, the Riemann Sum for the SigmaNotation F(w_i) Delta_iX is not unique.

I look forward to hearing back from you

Hi Mark,

To prove the integrability of F(x) from 0 to 1 you need to consider the limit

where |P| is the norm of the partition and w_i is a point in the interval from x_i-1 to x_i. This limit must exist and its value must NOT depend on the choice of w_i.

With the function that you have, no matter how small |P|, you can always find a rational and an irrational in each interval from x_i-1 to x_i. If you always choose w_i to be rational then the limit is 1, and if you always choose w_i to be irrational then the limit is 0. Thus the value of the limit depends on the choice of w_i and hence the integral does not exist.

I hope this helps,
Harley Go to Math Central