The fundamental theorem of calculus guarantees that if $f: \mathbb{R} \to \mathbb{R}$ is continuous, f has an antiderivative. But clearly f does not need to be continuous, since it is easy to construct functions which have discontinuous derivatives. Darboux's Theorem states that every real function with an antiderivative over a closed interval I satisfies the intermediate value property on I. But it is again not that difficult to prove that most functions with the intermediate value property (Darboux functions) do not have antiderivatives. Wikipedia claims every real function equals the sum of two Darboux functions, so they can't be closed under addition. Antidifferentiable functions certainly are closed under addition, so the two sets cannot be equal. Finally, Google supplies some existence theorems for antiderivatives of complex functions (a much stricter condition), but not for real functions.

Are there any strong conditions for the existence of an antiderivative of a real-valued function of real numbers? Certainly integrability is not enough. Is there even a word for these functions besides "antidifferentiable"?