# 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) .

**"Step-by-Step Explanation"**refers to a detailed and sequential breakdown of the solution or reasoning behind the answer. This comprehensive explanation walks through each step of the answer, offering you clarity and understanding.

Our explanations are based on the best information we have, but they may not always be right or fit every situation.

**blue check mark**means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.

Learn more on how we answer questions.

## Related Answered Questions

Question: 8.5

## Verified Answer:

Since the function f (x) = x − x^{2}[/latex...

Question: 8.T.16

## Verified Answer:

(i) Since f is decreasing and integrable over [k −...

Question: 8.T.15

## Verified Answer:

Corollary 8.10.3 ensures the existence of both int...

Question: 8.T.14

## Verified Answer:

Once again, we note that the range of φ is an inte...

Question: 8.T.13

## Verified Answer:

Let J be the interval whose end-points are φ (a) a...

Question: 8.T.12

## Verified Answer:

If P = \left\{x_{0}, x_{1}, ..., x_{n}\righ...

Question: 8.T.10

## Verified Answer:

Let ε > 0 be given. We shall prove the existenc...

Question: 8.T.9

## Verified Answer:

The case where f is constant on [a, b] can be dism...

Question: 8.T.8

## Verified Answer:

Suppose first that f ∈ \mathcal{R}(a, b) .[...

Question: 8.T.7

## Verified Answer:

Let P = \left\{x_{0}, x_{1}, ..., x_{n}\rig...