My question is, what does the image of the derivative look like?
I.e., if you differentiate an everywhere continuous function, what kinds of
functions do you get?
also, what if the original function is not everywhere differentiable; what
kind of partial functions do you get?

suppose $f$ is everywhere differentiable;
then look at $F=\int_0^x f'$
(this always makes sense if you use generalize Riemann integrals);
what's the difference between $f$ and $F$?
(conversely, given an IVT function, integrate then differentiate;
 what happens?? you can lose stuff at the discontinuity points -- what's the
 ``right''
value at these points?)
(perhaps in R. A. Gordon's book??)

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2nd question: I understand the image of the Lebesgue integral, both from $L^1$
(absolutely integrable) and from finite (variation) measures;
these give absolutely continuous functions and bounded variation functions.

What does the image of the Denjoy integral look like?
do we get all continous funtions?
all functions (if we integrate ``unbounded variation measures'')??
