Is there a formal definition for antiderivatives?

Question

In the way the derivative can be defined as a limit, specifically $$f'(x):=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ or any of the other possible variants, is there a way to define the antiderivative, as in indefinite integral?

The handful of sources I've looked over (Wikipedia and MathWorld, to name a few) all refer to the antiderivative simply as a "nonunique inverse operator" (I'm paraphrasing). I can't say I'm completely satisfied with this notion. Is the following the best we can do?

If $F(x)$ is a function that satisfies $\dfrac{d}{dx}F(x)=f(x)$, then $F(x)$ is called an antiderivative of $f(x)$. (Forgive the lack of formality.)

Is it possible to come up with a sort of "inverse-limit" to describe antiderivatives?

Answer 1

Since the antiderivative is so far from unique, we can't give anything like a pointwise formula for it as it looks like you want without choosing one. Probably the best idea is the definite integral, $$F(x)=\int_0^x f(x)dx=\lim_{n\to \infty} \frac{x}{n}\left(\sum_{k=0}^{n-1} f(kx/n)\right)$$ Here I've used the left Riemann sum, which is of course not the only way to define an integral, but it works whenever the integral exists. It's possible to write a less arbitrary definition, but uglier.

Answer 2

For all regulated functions you can define a sequence of step functions that converge to the function you wish to integrate. Then compute the Riemann sum over the partition.