Darboux's Theorem: If f is differentiable on [a,b], then f' has the intermediate value property: for any k between f'(a) and f'(b), there exists c in (a,b) with f'(c) = k.

Key implication 1: Derivatives cannot have jump discontinuities. If f' jumps from value A to value B at point p, it must take every value between A and B — but a jump discontinuity skips values. Contradiction.

Key implication 2: Not every function is a derivative. The sign function sgn(x) has a jump at 0, so it's not the derivative of any function. Similarly, [x] and step functions.

Key implication 3: If f' is known to exist but be discontinuous, the discontinuity must be oscillatory (like cos$\frac{1}{x}$ oscillating without settling).

Example: f(x) = $x^2$ sin$\frac{1}{x}$, f(0) = 0. f'(0) = 0, but for x != 0, f'(x) = 2x sin$\frac{1}{x}$ - cos$\frac{1}{x}$, which oscillates near 0. f' is discontinuous at 0, but the discontinuity is oscillatory — consistent with Darboux.

JEE application: "Given that f'(a) = 3 and f'(b) = -1, prove f'(c) = 0 for some c in (a,b)." Direct application of Darboux: 0 is between -1 and 3.