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Gao Wei

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Fundamental Theorem of Calculus
Let f:[a,b]→Rf:[a,b] \to \R be Riemann integrable. Let F:[a,b]→RF:[a,b]\to\R be F(x)=∫axf(t)dtF(x)= \int_{a}^{x}f(t)dt. Then FF is continuous, and at all xx such that ff is continuous at xx, FF is differentiable at xx with F′(x)=f(x)F'(x)=f(x).