Theorem: Fundamental Theorem of Calculus

Suppose f is an integrable function defined on the closed, bounded interval [a, b], define a new function:

F(x) = f(t) dt
Then F is continuous in [a, b]. Moreover, if f is also continous, then F is differentiable in (a, b) and

F'(x) = f(x) for all x in [a, b].

Note: In many calculus texts this theorem is called the Second fundamental theorem of calculus. Those books refer to this theorem as the first fundamental theorem of calculus

Proof: later

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