Is Thomae's function Riemann integrable on every interval?

Question

https://en.wikipedia.org/wiki/Thomae%27s_function

It seems to me that on every interval in $\mathbb{R}$, the function is bounded and its set of discontinuities is just the set of rational numbers in that interval, which is countable and therefore of measure zero, so it's Riemann integrable on every interval. But every page on the Internet I've found says that it's Riemann integrable on $[0,1]$ without saying anything about the rest of $\mathbb{R}$. Is my reasoning accurate?

Follow-up question: is it correct then to say that $F(x) = \int_0^x f(t) \,\mathrm{d}t$ where $f(t)$ denotes Thomae's function is by construction differentiable everywhere?

The motivation behind this is that I'm trying to look for a simple function that is differentiable but has a derivative that is discontinuous at infinitely many points.

Yes, this function is Riemann integrable on every interval. As for the second question, you can easily see that $F(x)$ is always zero, so $F'(x)$ is zero. — Crostul, Mar 07 '15 at 00:18
@Crostul: Hm I see why that's true but why isn't it valid to say that $F'(x) = f(x) - f(0) = f(x)$? — Marcus Emilsson, Mar 07 '15 at 00:20
@Crostul: Never mind I see that the FT of calculus requires continuity. — Marcus Emilsson, Mar 07 '15 at 00:21
Regarding your final paragraph, have you seen the "standard example" of a differentiable function with discontinuous derivative? — Andrew D. Hwang, Mar 07 '15 at 00:29
@user86418: I think so; I just read http://math.stackexchange.com/questions/292275/discontinuous-derivative/. — Marcus Emilsson, Mar 07 '15 at 00:31

Is Thomae's function Riemann integrable on every interval?

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