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Math Help - Existence of the Anti-Derivative.

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    Existence of the Anti-Derivative.

    Given a function f(x) defined on interval [a,b]. Prove that there must exist a function F(x) with the property that F'(x)=f(x) \forall x\in (a,b).

    With this we can prove the second fundamental theorem of calculus. We have to show that if g(x)=\int^x_a f(x)dx then g'(x)=f(x). 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 f(x) then \int^x_a f(x)dx = F(x)-F(a)" But by the existence of anti-derivative conjecture there MUST exist such a function thus, g(x)=F(x)-F(a) but then g'(x)=f(x) because F(a) is a constant-function.
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    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.
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