Darboux's Theorem: If f is differentiable on [a,b] and k is between f'(a) and f'(b), then f'(c) = k for some c in (a,b).

Key implication: Derivatives have the intermediate value property even if they are not continuous. This means f' cannot have jump discontinuities.

Example: f(x) = $x^2$ sin$\frac{1}{x}$, f(0) = 0. f'(0) = 0, but f'(x) = 2x sin$\frac{1}{x}$ - cos$\frac{1}{x}$ oscillates near 0. The discontinuity of f' at 0 is oscillatory, not a jump — consistent with Darboux.

JEE application: If a function g is defined as a derivative g = f' on some interval, and g has a jump discontinuity, then g cannot be a derivative — contradiction. This can be used to prove certain functions are not derivatives of any function.