final update on 4121

pages/Math4121/Exam_reviews/Math4121_E2.md

@@ -44,6 +44,45 @@ where $m_i = \inf_{x \in [x_{i-1}, x_i]} f(x)$ and $\alpha_i = \inf_{x \in [x_{i
 
 ### Fail of Riemann-Stieltjes Integration
 
+Consider the function
+
+$$
+((x)) = \begin{cases} 
+x-\lfloor x \rfloor & x \in [\lfloor x \rfloor, \lfloor x \rfloor + \frac{1}{2}) \\
+0 & x=\lfloor x \rfloor + \frac{1}{2}\\
+x-\lfloor x \rfloor - 1 & x \in (\lfloor x \rfloor + \frac{1}{2}, \lfloor x \rfloor + 1] \end{cases}
+$$
+
+![Graph of y=((x))](https://notenextra.trance-0.com/Math4121/y=((x)).png)
+
+We define
+
+$$
+f(x) = \sum_{n=1}^{\infty} \frac{((nx))}{n^2}=\lim_{N\to\infty}\sum_{n=1}^{N} \frac{((nx))}{n^2}
+$$
+
+![Graph of y=f(x)](https://notenextra.trance-0.com/Math4121/sum_y=((x)).png)
+
+(i) The series converges uniformly over $x\in[0,1]$.
+
+$$
+\left|f(x)-\sum_{n=1}^{N} \frac{((nx))}{n^2}\right|\leq \sum_{n=N+1}^{\infty}\frac{|((nx))|}{n^2}\leq \sum_{n=N+1}^{\infty} \frac{1}{n^2}<\epsilon
+$$
+
+As a consequence, $f(x)\in \mathscr{R}$.
+
+(ii) $f$ has a discontinuity at every rational number with even denominator.
+
+$$
+\begin{aligned}
+\lim_{h\to 0^+}f(\frac{a}{2b}+h)-f(\frac{a}{2b})&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))}{n^2}-\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}))}{n^2}\\
+&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
+&=\sum_{n=1}^{\infty}\lim_{h\to 0^+}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
+&>0
+\end{aligned}
+$$
+
+
 #### Some integrable functions are not differentiable (violates the fundamental theorem of calculus)
 
 Solve:
@@ -70,7 +109,7 @@ And we claim that the function is integrable on $[a,b]$ if and only if the outer
 
 > Outer content:
 >
-> The **outer content** of a set $S$ is the infimum of the lengths of all finite covers of $S$. $c_e(S) = \inf_{C\in \mathcal{C}_S}\ell(C)$. (e denotes "exterior")
+> The **outer content** of a set $S$ is the infimum of the lengths of all **finite covers** of $S$. $c_e(S) = \inf_{C\in \mathcal{C}_S}\ell(C)$. (e denotes "exterior")
 
 Homework question: You cannot cover an interval $[a,b]$ with length $k$ with a finite cover of length strictly less than $k$.
 
@@ -150,7 +189,7 @@ $\mathbb{R}$ is not first species.
 >
 > The **boundary** of a set $S$ is the set of all points in $S$ that are not in the interior of $S$. $\partial S = \overline{S} \setminus S^\circ$.
 
-#### Missing Thoerem 3.4
+#### Theorem 3.4
 
 Bolzano-Weierstrass Theorem:
 
@@ -190,12 +229,51 @@ For any open cover of a compact set, there exists a finite subcover.
 
 A set $S$ is **nowhere dense** if there are no open intervals in which $S$ is dense.
 
-That is equivalent to $S'$ contains no open intervals.
+That is equivalent to **$S'$ contains no open intervals**.
+
+Note: If $S$ is nowhere dense, then $S^c$ is dense. But if $S$ is dense, $S^c$ is not necessarily nowhere dense. (Consider $\mathbb{Q}$)
 
 ### Perfect Set
 
 A set $S$ is **perfect** if $S'=S$.
 
+Example: open intervals, Cantor set.
+
+#### Cantor set
+
+The Cantor set ($SVC(3)$) is the set of all real numbers in $[0,1]$ that can be represented in base 3 using only the digits 0 and 2.
+
+The outer content of the Cantor set is 0.
+
+#### Generalized Cantor set (SVC(n))
+
+The outer content of $SVC(n)$ is $\frac{n-3}{n-2}$.
+
+#### Lemma 4.4
+
+Osgood's Lemma:
+
+Let $G$ be a closed, bounded set and Let $G_1\subseteq G_2\subseteq \ldots$ and $G=\bigcup_{n=1}^{\infty} G_n$. Then $\lim_{n\to\infty} c_e(G_n)=c_e(G)$.
+
+Key: Using Heine-Borel Theorem.
+
+#### Theorem 4.5
+
+Arzela-Osgood Theorem:
+
+Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous, uniformly bounded functions on $[0,1]$ that converges pointwise to $0$. It follows that 
+
+$$
+\lim_{n\to\infty}\int_0^1 f_n(x) \, dx = \int_0^1 \lim_{n\to\infty} f_n(x) \, dx=0
+$$
+
+Key: Using Osgood's Lemma and do case analysis on bounded and unbounded parts of the Riemann-Stieltjes integral.
+
+#### Theorem 4.7
+
+Baire Category Theorem:
+
+An open interval cannot be covered by a countable union of nowhere dense sets.