Analysis Henstock-Kurzweil Integral

Question

a) Please help in proving that if $f$ is Riemann integrable on $[a,b]$ for some $a \lt b$, then $f$ is Henstock-Kurzweil integrable on $[a,b]$ with Henstock-Kurzweil integral equal to the Riemann integral of $f$, from $a$ to $b$.

b) Please show that $1_{\mathbb{Q} \cap [a,b]}$ is Henstock-Kurzweil integrable on $[a,b]$ for all $a \lt b$.

Can someone please explain the problem? I'm not understanding the hint.

Answer 1

Hint for (b). Let $\epsilon>0$ and define a gauge $\Delta$ on $[a,b]$ by taking $\Delta(r_k) = \epsilon/2^k$ where $r_k$ is the $k$th rational and $\Delta(x)=1$ for irrational $x$. Then show any $\Delta$-fine Riemann sum is small.