In article <email@example.com>, Kenshin <firstname.lastname@example.org> wrote:
> Let f : [a,b] -> R be differentiable everywhere and f ' be the > derivative of f. > > I've heard that > > 1) f ' may not be Riemann integrable. > > 2) f ' may not be even Lebesgue integrable. > > Can you give an example of this?
1) Correct. For one thing, f' need not be bounded on [a,b] (hence f' is not RI on [a,b]). Example: f(x) = x^2*sin(1/x^2) on [0,1]. Even if f' is bounded on [a,b], it need not be RI there. Recall that a bounded function on [a,b] is RI iff the set of its discontinuities has measure 0. But one can construct a differentiable f on [a,b] with f' bounded, but with f' discontinuous on a set of large measure.
2) f' will be Lebesgue measurable, certainly. But we can have int_a^b |f'(x)| dx = oo. In fact f(x) = x^2*sin(1/x^2) has this property. More exotic counterexamples can be constructed here.