Let f [a, b] → R be Riemann integrable (and in particular, bounded on [a, b]). For x = [a, b], let F(x) = f(t)dt. (i) Prove that F is continuous on [a, b].
Let f [a, b] → R be Riemann integrable (and in particular, bounded on [a, b]). For x = [a, b], let F(x) = f(t)dt. (i) Prove that F is continuous on [a, b].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( f: [a, b] \to \mathbb{R} \) be Riemann integrable (and in particular, bounded on \([a, b]\)). For \( x \in [a, b] \), let
\[
F(x) = \int_{a}^{x} f(t) \, dt.
\]
(i) Prove that \( F \) is continuous on \([a, b]\).
(ii) If \( f \) is continuous at \( x_0 \), prove that \( F \) is differentiable at \( x_0 \), and \( F'(x_0) = f(x_0) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F8d02e858-6c77-48e2-bd73-d9dc660201ac%2Fzw3xvv7_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f: [a, b] \to \mathbb{R} \) be Riemann integrable (and in particular, bounded on \([a, b]\)). For \( x \in [a, b] \), let
\[
F(x) = \int_{a}^{x} f(t) \, dt.
\]
(i) Prove that \( F \) is continuous on \([a, b]\).
(ii) If \( f \) is continuous at \( x_0 \), prove that \( F \) is differentiable at \( x_0 \), and \( F'(x_0) = f(x_0) \).
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(ii) please!
![Let ƒ : [a, b] → R be Riemann integrable (and in particular, bounded on [a, b]).
For x = [a, b], let F(x) = f f(t)dt.
(i) Prove that F is continuous on [a, b].
(ii) If f is continuous at xo, prove that F is differentiable at xo, and F'(xo)
f(xo).
=](https://content.bartleby.com/qna-images/question/e15ed467-90ec-4e60-afef-3d3f6119f74d/4e92873c-84f1-440d-b0aa-afc657a89e69/76jdyti_processed.png)
Transcribed Image Text:Let ƒ : [a, b] → R be Riemann integrable (and in particular, bounded on [a, b]).
For x = [a, b], let F(x) = f f(t)dt.
(i) Prove that F is continuous on [a, b].
(ii) If f is continuous at xo, prove that F is differentiable at xo, and F'(xo)
f(xo).
=
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