We define Riemann function
$ f\left(x\right)=\begin{cases} \frac{1}{q} & x=\frac{p}{q}\in\mathbb{Q},\left(\frac{p}{q}\text{ reduced}\right)\\ 0 & x\notin\mathbb{Q} \end{cases} $
and Dirichlet function
$ D\left(x\right)=\begin{cases} 1 & x\in\mathbb{Q}\\ 0 & x\notin\mathbb{Q} \end{cases} $
I have to determine which of the following is integrable over $ [0,1] $ :
$$1.D(x)f(x)\ \ 2. D(\sqrt{2}x)\ \ 3. f(\sqrt{2}x)\ \ 4.D(\sqrt{2}x)f(x) $$
Here's what I tried:
$1$. I know that Riemann function is integrable in $ [0,1] $ and basically $ D(x)f(x)= f(x) $ in $ [0,1] $ so the first one is integrable.
$2$. I claim that $ D\left(\sqrt{2}x\right) $ is not integrable, because if it was then we would get that $ D(x) $ is integrable in $ [0,\sqrt{2}] $ and :
$ \intop_{0}^{1}D\left(\sqrt{2}x\right)=\frac{1}{\sqrt{2}}\intop_{0}^{\sqrt{2}}D\left(x\right) $
But it's false. so the second is not integrable.
As for 3, I think that its integrable, because $ f $ is integrable in $ [0,\sqrt{2}] $ (is it? ) therefore $ f(\sqrt{2}x) $ is integrable in $ [0,1] $
And for the last one, I cannot determine.
I'd appreciate any kind of help. Thanks in advance