A function that is not a derivative If this is possible, could somebody give me an example of a function $f:[a,b]\to \Bbb{R}$ continuous in $[a,b]$ but that THERE IS NOT a function, namely G, such that its derivative is the funciont f? i.e., such that
$$G'=f.$$
This isn't possible: Every such function is a derivative. For any continuous function $f: [a, b] \to \mathbb{R}$, we can define the function
$$G(x) := \int_a^x f(t) \,dt,$$
and by the Fundamental Theorem of Calculus,
$$G'(x) = f(x).$$
(This assumes that we define the derivative of a function at the endpoint of an interval to be the value of the appropriate one-sided limit.)
I'm not pretty sure but this could be what you are looking for:

http://en.wikipedia.org/wiki/Weierstrass_function