Elegant Meaning of an Integral

For the formalists out there I have been troubled by the meaning of,

Because what is that supposed to represent?

It is more appropriate to think of an integral as a **set** of functions rather than a function.

So here is my version, tell me if you like it.

Let represent the set of all real functions that are differenciable on some open interval. Let represent the set of all functions. *)

We can view the sets and as groups under functional addition.

Then we can define the *derivative* as a group homomorphism,

I think the beauty in this is defining what an *integral* is. Given any function we define,

*)Since I am not a set theorist (noob :mad: ) I cannot tell whether such a set can exist.

Note: It was not necessary for me to mention a group homomorphism I just was thinking whether the kernel of the derivative will be of any use. I do not see of one.