From f78c5149f93ebc7a123c5c4759f2bf03d0554ebb Mon Sep 17 00:00:00 2001 From: stratisMarkou Date: Wed, 1 May 2024 11:43:11 +0300 Subject: [PATCH] Update documentation --- _sources/book/mira/000-exercises.md | 59 +++++++++++++++++++++++++++++ _sources/book/mira/001-riemann.md | 1 + book/mira/000-exercises.html | 57 ++++++++++++++++++++++++++++ book/mira/001-riemann.html | 2 +- prf-prf.html | 29 +++++++++----- searchindex.js | 2 +- 6 files changed, 138 insertions(+), 12 deletions(-) diff --git a/_sources/book/mira/000-exercises.md b/_sources/book/mira/000-exercises.md index 29a792f1..1bbb557c 100644 --- a/_sources/book/mira/000-exercises.md +++ b/_sources/book/mira/000-exercises.md @@ -473,6 +473,65 @@ Since $\epsilon$ can be made arbitrarily small, it follows that $f$ is Riemann i +::::{admonition} Exercise 1.A.11 +:class: tip + +Suppose $f: [a, b] \to \mathbb{R}$ is Riemann integrable. +Define $F: [a, b] \to \mathbb{R}$ by + +$$F(t) = \begin{cases} +0 & \text{ if } t = a \\ +\int_a^t & \text{ if } t \in (a, b]. +\end{cases}$$ + +Prove that $F$ is continuous on $[a, b].$ + +:::{dropdown} Solution +Let $t_0 \in [a, b]$ and $\epsilon > 0.$ +Since $f$ is Riemann integrable, it {prf:ref}`is boundedd` by some $C \in \mathbb{R}.$ +For any $\delta > 0$ and $x \in [a, b],$ if $|t - t_0| < \delta,$ then + +$$|F(t) - F(t_0)| = \left|\int^t_{t_0} f\right| < \delta C.$$ + +Therefore, by picking $\delta < \epsilon / C$ we have $|F(t) - F(t_0)| < \epsilon,$ showing that $F$ is continuous. +::: +:::: + + +::::{admonition} Exercise 1.A.12 +:class: tip + +Suppose $f: [a, b] \to \mathbb{R}$ is Riemann integrable. +Prove taht $|f|$ is Riemann integrable and that + +$$\left|\int_a^b f\right| \leq \int_a^b |f|.$$ + +:::{dropdown} Solution +Let $\epsilon > 0.$ +Since $f$ is Riemann integrable, there exists a partition $P_\epsilon = (x_0, x_1, \dots, x_n)$ of $[a, b]$ such that + +$$U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) < \epsilon.$$ + +Now since + +$$\begin{align} +\sup_{y \in [x_{k-1}, x_k]} |f(y)| - \inf_{z \in [x_{k-1}, x_k]} |f(z)| &\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\ +&\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\ +\end{align}$$ + +we have that + +$$U(|f|, P_\epsilon, [a, b]) - L(|f|, P_\epsilon, [a, b]) \leq U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) < \epsilon$$ + +so $|f|$ is Riemann integrable. +Since $U(|f|, P_\epsilon, [a, b]) \geq U(f, P_\epsilon, [a, b]),$ we have that + +$$\left|\int_a^b f\right| \leq \int_a^b |f|.$$ +::: +:::: + + + ## Chapter 2.C ::::{admonition} Exercise 2.C.1 diff --git a/_sources/book/mira/001-riemann.md b/_sources/book/mira/001-riemann.md index 4bb2f55e..360441ae 100644 --- a/_sources/book/mira/001-riemann.md +++ b/_sources/book/mira/001-riemann.md @@ -220,6 +220,7 @@ Since this holds for any $\epsilon > 0,$ we have $U(f, [a, b]) = L(f, [a, b])$ a The last result in this section concerns a frequently used estimate for the Riemann integral of a function. :::{prf:theorem} Bounds on the Riemann integral +:label: mira-bounds-on-riemann-integral Suppose $f: [a, b] \to \mathbb{R}$ is Riemann integrable. Then diff --git a/book/mira/000-exercises.html b/book/mira/000-exercises.html index 3055a5c1..875fe180 100644 --- a/book/mira/000-exercises.html +++ b/book/mira/000-exercises.html @@ -846,6 +846,63 @@

Chapter 1.ASince \(\epsilon\) can be made arbitrarily small, it follows that \(f\) is Riemann integrable on \([a, b].\)

+
+

Exercise 1.A.11

+

Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. +Define \(F: [a, b] \to \mathbb{R}\) by

+
+\[\begin{split}F(t) = \begin{cases} +0 & \text{ if } t = a \\ +\int_a^t & \text{ if } t \in (a, b]. +\end{cases}\end{split}\]
+

Prove that \(F\) is continuous on \([a, b].\)

+
+ +Solution
+
+
+
+
+

Let \(t_0 \in [a, b]\) and \(\epsilon > 0.\) +Since \(f\) is Riemann integrable, it is boundedd by some \(C \in \mathbb{R}.\) +For any \(\delta > 0\) and \(x \in [a, b],\) if \(|t - t_0| < \delta,\) then

+
+\[|F(t) - F(t_0)| = \left|\int^t_{t_0} f\right| < \delta C.\]
+

Therefore, by picking \(\delta < \epsilon / C\) we have \(|F(t) - F(t_0)| < \epsilon,\) showing that \(F\) is continuous.

+
+
+
+

Exercise 1.A.12

+

Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. +Prove taht \(|f|\) is Riemann integrable and that

+
+\[\left|\int_a^b f\right| \leq \int_a^b |f|.\]
+
+ +Solution
+
+
+
+
+

Let \(\epsilon > 0.\) +Since \(f\) is Riemann integrable, there exists a partition \(P_\epsilon = (x_0, x_1, \dots, x_n)\) of \([a, b]\) such that

+
+\[U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) < \epsilon.\]
+

Now since

+
+\[\begin{split}\begin{align} +\sup_{y \in [x_{k-1}, x_k]} |f(y)| - \inf_{z \in [x_{k-1}, x_k]} |f(z)| &\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\ +&\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\ +\end{align}\end{split}\]
+

we have that

+
+\[U(|f|, P_\epsilon, [a, b]) - L(|f|, P_\epsilon, [a, b]) \leq U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) < \epsilon\]
+

so \(|f|\) is Riemann integrable. +Since \(U(|f|, P_\epsilon, [a, b]) \geq U(f, P_\epsilon, [a, b]),\) we have that

+
+\[\left|\int_a^b f\right| \leq \int_a^b |f|.\]
+
+

Chapter 2.C#

diff --git a/book/mira/001-riemann.html b/book/mira/001-riemann.html index 64ea6b72..7019c636 100644 --- a/book/mira/001-riemann.html +++ b/book/mira/001-riemann.html @@ -622,7 +622,7 @@

Riemann integral\(\epsilon > 0,\) we have \(U(f, [a, b]) = L(f, [a, b])\) and so \(f\) is Riemann integrable.

The last result in this section concerns a frequently used estimate for the Riemann integral of a function.

-
+

Theorem 58 (Bounds on the Riemann integral)

Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. diff --git a/prf-prf.html b/prf-prf.html index 5724fcf0..47788b96 100644 --- a/prf-prf.html +++ b/prf-prf.html @@ -423,6 +423,7 @@

Proof Index

lemma-2 | lemma-3 | lemma-8 | + mira-bounds-on-riemann-integral | mira-eg-not-riemann-integrable | mira-thm-condition-measurable | mira-thm-measure-increasing-union | @@ -497,7 +498,7 @@

Proof Index

- definition-0 (book/toc/002-cfl) + definition-0 (book/mira/001-riemann) definition   @@ -505,7 +506,7 @@

Proof Index

- definition-1 (book/toc/002-cfl) + definition-1 (book/mira/001-riemann) definition   @@ -561,7 +562,7 @@

Proof Index

- definition-2 (book/toc/002-cfl) + definition-2 (book/mira/001-riemann) definition   @@ -633,7 +634,7 @@

Proof Index

- definition-5 (book/toc/002-cfl) + definition-5 (book/mira/001-riemann) definition   @@ -649,7 +650,7 @@

Proof Index

- definition-7 (book/toc/002-cfl) + definition-7 (book/mira/001-riemann) definition   @@ -681,7 +682,7 @@

Proof Index

- example-12 (book/toc/002-cfl) + example-12 (book/mira/001-riemann) example   @@ -804,6 +805,14 @@

Proof Index

lemma-8 (book/toc/002-cfl) lemma   + + mira-bounds-on-riemann-integral + + + + mira-bounds-on-riemann-integral (book/mira/001-riemann) + theorem +   mira-eg-not-riemann-integrable @@ -985,7 +994,7 @@

Proof Index

- theorem-14 (book/toc/001-fsa) + theorem-14 (book/mira/001-riemann) theorem   @@ -1113,7 +1122,7 @@

Proof Index

- theorem-4 (book/prob-intro/ch08/content) + theorem-4 (book/mira/001-riemann) theorem   @@ -1137,7 +1146,7 @@

Proof Index

- theorem-6 (book/prob-intro/ch08/content) + theorem-6 (book/mira/001-riemann) theorem   @@ -1153,7 +1162,7 @@

Proof Index

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