I can see that no one posed an answer. It is indeed a difficult problem. It is true forcontinousfunctions 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 FunctionDOES NOThave an anti-derivative.