True or false that all real function is an antiderivative

[kallazans]

is it true or false that all real function is an antiderivative of some real function but neither real function have an antiderivative?

I still have the doubt!

Definition(Louis Leithold,The Calculus with Analytic Geometry)
Antiderivative: F is antiderivative of f in I if F'(x)=f(x) for all x in I.

The question is all f have some F in some I?
The question is all f is a G of some g in some I?
(Real Analysis)

 

[Hurkyl]

By "primitive" do you mean that if $f(x) = \int g(x) \, dx$ then $f(x)$ is a primitive of $g(x)$? (The usual English word for this is that $f(x)$ is an antiderivative or an integral of $g(x)$)


I'm not entirely sure what you're trying to ask... though it is false that any function is an antiderivative of another function.

 

[HallsofIvy]

It's not clear what you mean by "either" function. If you mean the original function in the question and its anti-derivative, then obviously IF "every function had a anti-derivative", then it wouldn't make sense to say that THAT function did NOT have an anti-derivative.
However, as Hurkyl pointed out, it is not true that every function has a primitive (anti-derivative). For example, the function, f(x)= 1 if x is rational, 0 if x is irrational, does not have an antiderivative.
It IS true that every bounded function whose points of discontinuity form a set of measure 0 is integrable (has an anti-derivative). In particular every continuous function has an anti-derivative as well as every bounded function with only a finite number of points of discontinuity.