# Thread: show function is not integrable

1. ## show function is not integrable

Let [0,1] ----> R be given by f(x) = 1 if x is rational, -1 if x is rational.

Decide whether f is integrable.

I know the function is not integrable, but I don't have any idea how to go about showing it? Been looking through some Riemann integral lecture notes but don't understand it at all!

2. Originally Posted by hunkydory19
Let [0,1] ----> R be given by f(x) = 1 if x is rational, -1 if x is rational.

Decide whether f is integrable.
Let $\displaystyle P = \{x_0 < x_1 < ... <x_n]$ be a partition. Then the upper Darboux sum of $\displaystyle f$ with respect to $\displaystyle P$ is $\displaystyle 1$ (why?). Thus, the infimum of all upper Darboux sums is $\displaystyle 1$. While the lower Darboux sum of $\displaystyle f$ with respect to $\displaystyle P$ is $\displaystyle 0$ (why?). Thus, the supremum of all lower Darboux sums is $\displaystyle 0$. This means the upper integral does not agree with the lower integral. Thus, the (Dirichlet) function is not integrable on $\displaystyle [0,1]$.

3. Thanks so much for the reply, but for the lower Darboux sum I don't understand why it is 0 and not -1...

4. Originally Posted by hunkydory19
Thanks so much for the reply, but for the lower Darboux sum I don't understand why it is 0 and not -1...
It is -1, I made a mistake.