Are derivatives integrable?

If $f$ is everywhere differentiable, is $f'$ integrable?

In terms of indefinite integral, yes, clearly: it's just algebra, and
$f$ works.

Further, it certainly has an integral over any -interval-, via
$\int_a^b f' = f(b)-f(a)$.
OTOH, it needn't have an integral over every -measurable set-:
it needn't be $L^1$, for instance.
(it can be ``conditionally integrable'', like
 $(\sin x)/x$ or $[(1/x^2)\sin (1/x^3)]'$)

However, one -can- define good integrals for these, and the FTC works
for them.
That is, virtually any theorem/proof that says ``if $f'$ is integrable''
works for any $f$ that's everywhere differentiable,
and I hold that these are The Right proofs to give
(or at least very useful ones).
[...the alternative is usually to use the MVT and give an ugly proof]
...as this captures the ideal, the morality, and generalizes as soon
as you make a better integral.
