Darboux's theorem
You might expect a function that is itself a derivative to be reasonably tame. Surprisingly, a derivative can be discontinuous — yet it can never jump. Darboux's theorem reveals that derivatives, however wild, always honor the intermediate value property: they cannot skip values, even when they fail to be continuous.
Statement: if f is differentiable on an interval [a, b], then the derivative f' takes every value between f'(a) and f'(b). That is, for any y strictly between f'(a) and f'(b), there exists c in (a, b) with f'(c) = y. The remarkable part is that this holds with no assumption that f' is continuous — the intermediate value theorem normally requires continuity, but derivatives get this property for free.
The consequence is a sharp structural restriction: a derivative cannot have a jump discontinuity. So a function like the step function (which jumps) is not the derivative of anything. This explains why a derivative's only possible discontinuities are of the wilder, oscillatory kind (essential discontinuities), as in the derivative of x^2 sin(1/x). The proof considers g(x) = f(x) - yx, which has an interior extremum, then applies Fermat's theorem to conclude g'(c) = 0.
g(x) = x^2 sin(1/x) (with g(0) = 0) has derivative g'(x) = 2x sin(1/x) - cos(1/x) for x not 0 and g'(0) = 0; g' is discontinuous at 0 (it oscillates), yet by Darboux it still has no jump.
A derivative may be discontinuous, but its only discontinuities are oscillatory, never jumps.
Functions with the intermediate value property are called Darboux functions; the theorem says every derivative is a Darboux function, but not conversely — many Darboux functions are not derivatives.