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Question 8.T.11: Suppose that f ∈ R(a, b), and let F : [a, b] → R be defined ...

Suppose that f ∈ \mathcal{R}(a, b) , and let F : [a, b] → \mathbb{R} be defined by

F (x) = \int_{a}^{x}{f (t) dt}.

Then

(i) F is continuous and satisfies a Lipschitz condition on [a, b] .

(ii) If f is continuous at c ∈ [a, b] , F is differentiable at c and

F^{\prime} (c) = f (c) .

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