Thread: Existence of the Anti-Derivative.

Given a function defined on interval . Prove that there must exist a function with the property that .

With this we can prove the second fundamental theorem of calculus. We have to show that if then . Instead of the classical proof with a Riemann Sum we may do the following: Since by the first fundamental theorem of calculus, "If there exists an anti-derivative of then " But by the existence of anti-derivative conjecture there MUST exist such a function thus, but then because is a constant-function.

I can see that no one posed an answer. It is indeed a difficult problem. It is true for continous functions on a closed interval. Because this is a basic application of the second fundamental theorem of calculus (with the fact that countinous functions are Riemann integratble).

Now I was thinking in sake of a contradiction. Use the Dirichelt Function. Show that the Dirchelet Function DOES NOT have an anti-derivative.