Question

Do all functions have an antiderivative?
Do all functions have a derivative?

Answer 1

no, and no XD

Answer 2

No and no. For instance
f(x) = 1, if x is a rational number
= 0, if x is an irrational number

This function has neither a derivative nor an anti-derivative.

Answer 3

best function ever! XD

Answer 4

In fact, any function that isn't continuous at a point doesn't have a derivative at that point.

The set of functions that have anti-derivatives is interesting, but still quite restrictive.

Answer 5

[
It turns out there is a better notion of integration that can integrate the function I wrote down above. It would integrate it to zero, because if you chose a real number at random, call it x, it would be irrational with probability 1 and hence f(x) = 0.

This notion of integration is called the Lebesgue integral, and is studied in Real Analysis.
]