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59 changes: 59 additions & 0 deletions _sources/book/mira/000-exercises.md
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Expand Up @@ -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<mira-bounds-on-riemann-integral>` 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
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1 change: 1 addition & 0 deletions _sources/book/mira/001-riemann.md
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Expand Up @@ -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
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57 changes: 57 additions & 0 deletions book/mira/000-exercises.html
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Expand Up @@ -846,6 +846,63 @@ <h2>Chapter 1.A<a class="headerlink" href="#chapter-1-a" title="Link to this hea
<p class="sd-card-text">Since <span class="math notranslate nohighlight">\(\epsilon\)</span> can be made arbitrarily small, it follows that <span class="math notranslate nohighlight">\(f\)</span> is Riemann integrable on <span class="math notranslate nohighlight">\([a, b].\)</span></p>
</div>
</details></div>
<div class="tip admonition">
<p class="admonition-title">Exercise 1.A.11</p>
<p>Suppose <span class="math notranslate nohighlight">\(f: [a, b] \to \mathbb{R}\)</span> is Riemann integrable.
Define <span class="math notranslate nohighlight">\(F: [a, b] \to \mathbb{R}\)</span> by</p>
<div class="math notranslate nohighlight">
\[\begin{split}F(t) = \begin{cases}
0 &amp; \text{ if } t = a \\
\int_a^t &amp; \text{ if } t \in (a, b].
\end{cases}\end{split}\]</div>
<p>Prove that <span class="math notranslate nohighlight">\(F\)</span> is continuous on <span class="math notranslate nohighlight">\([a, b].\)</span></p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
<summary class="sd-summary-title sd-card-header">
Solution<div class="sd-summary-down docutils">
<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-down" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M5.22 8.72a.75.75 0 000 1.06l6.25 6.25a.75.75 0 001.06 0l6.25-6.25a.75.75 0 00-1.06-1.06L12 14.44 6.28 8.72a.75.75 0 00-1.06 0z"></path></svg></div>
<div class="sd-summary-up docutils">
<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-up" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M18.78 15.28a.75.75 0 000-1.06l-6.25-6.25a.75.75 0 00-1.06 0l-6.25 6.25a.75.75 0 101.06 1.06L12 9.56l5.72 5.72a.75.75 0 001.06 0z"></path></svg></div>
</summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">Let <span class="math notranslate nohighlight">\(t_0 \in [a, b]\)</span> and <span class="math notranslate nohighlight">\(\epsilon &gt; 0.\)</span>
Since <span class="math notranslate nohighlight">\(f\)</span> is Riemann integrable, it <a class="reference internal" href="001-riemann.html#mira-bounds-on-riemann-integral">is boundedd</a> by some <span class="math notranslate nohighlight">\(C \in \mathbb{R}.\)</span>
For any <span class="math notranslate nohighlight">\(\delta &gt; 0\)</span> and <span class="math notranslate nohighlight">\(x \in [a, b],\)</span> if <span class="math notranslate nohighlight">\(|t - t_0| &lt; \delta,\)</span> then</p>
<div class="math notranslate nohighlight">
\[|F(t) - F(t_0)| = \left|\int^t_{t_0} f\right| &lt; \delta C.\]</div>
<p class="sd-card-text">Therefore, by picking <span class="math notranslate nohighlight">\(\delta &lt; \epsilon / C\)</span> we have <span class="math notranslate nohighlight">\(|F(t) - F(t_0)| &lt; \epsilon,\)</span> showing that <span class="math notranslate nohighlight">\(F\)</span> is continuous.</p>
</div>
</details></div>
<div class="tip admonition">
<p class="admonition-title">Exercise 1.A.12</p>
<p>Suppose <span class="math notranslate nohighlight">\(f: [a, b] \to \mathbb{R}\)</span> is Riemann integrable.
Prove taht <span class="math notranslate nohighlight">\(|f|\)</span> is Riemann integrable and that</p>
<div class="math notranslate nohighlight">
\[\left|\int_a^b f\right| \leq \int_a^b |f|.\]</div>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
<summary class="sd-summary-title sd-card-header">
Solution<div class="sd-summary-down docutils">
<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-down" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M5.22 8.72a.75.75 0 000 1.06l6.25 6.25a.75.75 0 001.06 0l6.25-6.25a.75.75 0 00-1.06-1.06L12 14.44 6.28 8.72a.75.75 0 00-1.06 0z"></path></svg></div>
<div class="sd-summary-up docutils">
<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-up" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M18.78 15.28a.75.75 0 000-1.06l-6.25-6.25a.75.75 0 00-1.06 0l-6.25 6.25a.75.75 0 101.06 1.06L12 9.56l5.72 5.72a.75.75 0 001.06 0z"></path></svg></div>
</summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">Let <span class="math notranslate nohighlight">\(\epsilon &gt; 0.\)</span>
Since <span class="math notranslate nohighlight">\(f\)</span> is Riemann integrable, there exists a partition <span class="math notranslate nohighlight">\(P_\epsilon = (x_0, x_1, \dots, x_n)\)</span> of <span class="math notranslate nohighlight">\([a, b]\)</span> such that</p>
<div class="math notranslate nohighlight">
\[U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) &lt; \epsilon.\]</div>
<p class="sd-card-text">Now since</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{align}
\sup_{y \in [x_{k-1}, x_k]} |f(y)| - \inf_{z \in [x_{k-1}, x_k]} |f(z)| &amp;\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\
&amp;\leq \sup_{y \in [x_{k-1}, x_k]} f(y) - \inf_{z \in [x_{k-1}, x_k]} f(z) \\
\end{align}\end{split}\]</div>
<p class="sd-card-text">we have that</p>
<div class="math notranslate nohighlight">
\[U(|f|, P_\epsilon, [a, b]) - L(|f|, P_\epsilon, [a, b]) \leq U(f, P_\epsilon, [a, b]) - L(f, P_\epsilon, [a, b]) &lt; \epsilon\]</div>
<p class="sd-card-text">so <span class="math notranslate nohighlight">\(|f|\)</span> is Riemann integrable.
Since <span class="math notranslate nohighlight">\(U(|f|, P_\epsilon, [a, b]) \geq U(f, P_\epsilon, [a, b]),\)</span> we have that</p>
<div class="math notranslate nohighlight">
\[\left|\int_a^b f\right| \leq \int_a^b |f|.\]</div>
</div>
</details></div>
</section>
<section id="chapter-2-c">
<h2>Chapter 2.C<a class="headerlink" href="#chapter-2-c" title="Link to this heading">#</a></h2>
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2 changes: 1 addition & 1 deletion book/mira/001-riemann.html
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Expand Up @@ -622,7 +622,7 @@ <h2>Riemann integral<a class="headerlink" href="#riemann-integral" title="Link t
Since this holds for any <span class="math notranslate nohighlight">\(\epsilon &gt; 0,\)</span> we have <span class="math notranslate nohighlight">\(U(f, [a, b]) = L(f, [a, b])\)</span> and so <span class="math notranslate nohighlight">\(f\)</span> is Riemann integrable.</p>
</div>
</details><p>The last result in this section concerns a frequently used estimate for the Riemann integral of a function.</p>
<div class="proof theorem admonition" id="theorem-9">
<div class="proof theorem admonition" id="mira-bounds-on-riemann-integral">
<p class="admonition-title"><span class="caption-number">Theorem 58 </span> (Bounds on the Riemann integral)</p>
<section class="theorem-content" id="proof-content">
<p>Suppose <span class="math notranslate nohighlight">\(f: [a, b] \to \mathbb{R}\)</span> is Riemann integrable.
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29 changes: 19 additions & 10 deletions prf-prf.html
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Expand Up @@ -423,6 +423,7 @@ <h1>Proof Index</h1>
<a href="#cap-lemma-2"><strong>lemma-2</strong></a> |
<a href="#cap-lemma-3"><strong>lemma-3</strong></a> |
<a href="#cap-lemma-8"><strong>lemma-8</strong></a> |
<a href="#cap-mira-bounds-on-riemann-integral"><strong>mira-bounds-on-riemann-integral</strong></a> |
<a href="#cap-mira-eg-not-riemann-integrable"><strong>mira-eg-not-riemann-integrable</strong></a> |
<a href="#cap-mira-thm-condition-measurable"><strong>mira-thm-condition-measurable</strong></a> |
<a href="#cap-mira-thm-measure-increasing-union"><strong>mira-thm-measure-increasing-union</strong></a> |
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<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#definition-0"><code class="xref">definition-0</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#definition-0"><code class="xref">definition-0</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>definition</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-definition-1"><td></td><td>
<strong>definition-1</strong></td><td></td></tr>
<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#definition-1"><code class="xref">definition-1</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#definition-1"><code class="xref">definition-1</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>definition</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-definition-10"><td></td><td>
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<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#definition-2"><code class="xref">definition-2</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#definition-2"><code class="xref">definition-2</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>definition</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-definition-20"><td></td><td>
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<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#definition-5"><code class="xref">definition-5</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#definition-5"><code class="xref">definition-5</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>definition</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-definition-6"><td></td><td>
Expand All @@ -649,7 +650,7 @@ <h1>Proof Index</h1>
<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#definition-7"><code class="xref">definition-7</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#definition-7"><code class="xref">definition-7</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>definition</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-definition-8"><td></td><td>
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<tr>
<td></td>
<td>
<a href="book/toc/002-cfl.html#example-12"><code class="xref">example-12</code></a> <em>(book/toc/002-cfl)</em></td><td>
<a href="book/mira/001-riemann.html#example-12"><code class="xref">example-12</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>example</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-example-13"><td></td><td>
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<a href="book/toc/002-cfl.html#lemma-8"><code class="xref">lemma-8</code></a> <em>(book/toc/002-cfl)</em></td><td>
<em>lemma</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-mira-bounds-on-riemann-integral"><td></td><td>
<strong>mira-bounds-on-riemann-integral</strong></td><td></td></tr>
<tr>
<td></td>
<td>
<a href="book/mira/001-riemann.html#mira-bounds-on-riemann-integral"><code class="xref">mira-bounds-on-riemann-integral</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>theorem</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-mira-eg-not-riemann-integrable"><td></td><td>
<strong>mira-eg-not-riemann-integrable</strong></td><td></td></tr>
<tr>
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<tr>
<td></td>
<td>
<a href="book/toc/001-fsa.html#theorem-14"><code class="xref">theorem-14</code></a> <em>(book/toc/001-fsa)</em></td><td>
<a href="book/mira/001-riemann.html#theorem-14"><code class="xref">theorem-14</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>theorem</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-theorem-15"><td></td><td>
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<tr>
<td></td>
<td>
<a href="book/prob-intro/ch08/content.html#theorem-4"><code class="xref">theorem-4</code></a> <em>(book/prob-intro/ch08/content)</em></td><td>
<a href="book/mira/001-riemann.html#theorem-4"><code class="xref">theorem-4</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>theorem</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-theorem-40"><td></td><td>
Expand All @@ -1137,7 +1146,7 @@ <h1>Proof Index</h1>
<tr>
<td></td>
<td>
<a href="book/prob-intro/ch08/content.html#theorem-6"><code class="xref">theorem-6</code></a> <em>(book/prob-intro/ch08/content)</em></td><td>
<a href="book/mira/001-riemann.html#theorem-6"><code class="xref">theorem-6</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>theorem</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-theorem-7"><td></td><td>
Expand All @@ -1153,7 +1162,7 @@ <h1>Proof Index</h1>
<tr>
<td></td>
<td>
<a href="book/prob-intro/ch07/content.html#theorem-8"><code class="xref">theorem-8</code></a> <em>(book/prob-intro/ch07/content)</em></td><td>
<a href="book/mira/001-riemann.html#theorem-8"><code class="xref">theorem-8</code></a> <em>(book/mira/001-riemann)</em></td><td>
<em>theorem</em></td></tr>
<tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
<tr class="cap" id="cap-theorem-9"><td></td><td>
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