Henstock–Kurzweil integrability does not imply integrability of the absolute value

Given: The derivative $f$ from the stated example.

The derivative $f$ of $x^2\sin (1\mathrm{/}{x}^2)$ is Henstock–Kurzweil integrable on $[0,1]$ ($F(x)=x^2\sin (1\mathrm{/}{x}^2)$ has an unbounded derivative whose Henstock–Kurzweil integral is $\sin 1$).

A finite-endpoint noncompact integral extends to a proper HK integral if and only if the truncation limit exists (Hake's theorem: a finite-endpoint generalized integral is a proper Henstock–Kurzweil integral after assigning the endpoint value).

The harmonic series ${\sum }_{k\ge 1}1\mathrm{/}k$ diverges (For rational $p>0$, $\sum 1\mathrm{/}{k}^p$ converges iff $p>1$).

Substitution for derivatives evaluates the integral of a composed derivative by its endpoint values (Henstock–Kurzweil substitution for a derivative composed with a differentiable map).

If $p$ and $q$ are HK integrable on a compact interval and $p\le q$ there, then $\int p\le \int q$ (Monotonicity of the Henstock–Kurzweil integral).

Every continuous function on a compact interval is Riemann integrable (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

Every Riemann integrable function is Henstock–Kurzweil integrable with the same integral (Every Riemann integrable function is Henstock–Kurzweil integrable with the same integral).