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pages/Math4121/Math4121_L1.md

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-# Lecture 1
+# Math4121 Lecture 1
 
 ## Chapter 5: Differentiation
 

pages/Math4121/Math4121_L10.md

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-# Lecture 10
+# Math 4121 Lecture 10
 
 ## Recap
 

pages/Math4121/Math4121_L13.md

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-# Lecture 13
+# Math4121 Lecture 13
+
+## New book Chapter 2
+
+Riemann's motivation: Fourier series
+
+$$
+F(x) = \frac{a_0}{2} + \sum_{k=1}^{\infty} a_k \cos(kx) + b_k \sin(kx)
+$$
+
+$a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(kx) dx$
+
+$b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(kx) dx$
+
+To study the convergence of the Fourier series, we need to study the convergence of the sequence of partial sums. (Riemann integration)
+
+Why Riemann integration?
+
+Let
+
+$$
+((x)) = \begin{cases} 
+x-\lfloor x \rfloor & x \in [\lfloor x \rfloor, \lfloor x \rfloor + \frac{1}{2}) \\
+0 & x=\lfloor x \rfloor + \frac{1}{2}\\
+x-\lfloor x \rfloor - 1 & x \in (\lfloor x \rfloor + \frac{1}{2}, \lfloor x \rfloor + 1] \end{cases}
+$$
+
+We define
+
+$$
+f(x) = \sum_{n=1}^{\infty} \frac{((nx))}{n^2}=\lim_{N\to\infty}\sum_{n=1}^{N} \frac{((nx))}{n^2}
+$$
+
+(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.
+
+$$
+\begin{aligned}
+\lim_{h\to 0^+}f(\frac{a}{2b}+h)-f(\frac{a}{2b})&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))}{n^2}-\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}))}{n^2}\\
+&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
+&=\sum_{n=1}^{\infty}\lim_{h\to 0^+}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
+&>0
+\end{aligned}
+$$
+
+### Back to the fundamental theorem of calculus
+
+Suppose $f$ is integrable on $[a,b]$, then
+
+$$
+F(x)=\int_a^x f(t)dt
+$$
+
+$F$ is continuous on $[a,b]$.
+
+if $f$ is continuous at $x_0$, then $F$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$.
+
+#### Theorem (Darboux's theorem)
+
+If $\lim_{x\to a^-}f(x)=L^-$, then $\lim_{h\to 0} \frac{f(a+h)-f(a)}{h}=L^-$.
+
+Proof:
+
+$$
+h\sup_{x\in [0,h]}f(x)\geq F(a+h)\geq \inf_{x\in [0,h]}f(x)h
+$$
+
+Consequently,
+
+$$
+f(x)=\sum_{n=1}^{\infty} \frac{((nx))}{n^2}
+$$
+
+then
+
+$$
+F(x)=\int_0^x f(t)dt
+$$
+
+is continuous on $[0,1]$.
+
+However, since $\lim_{x\to 0^+}f(x)\neq \lim_{x\to 0^-}f(x)$ holds for all the rational numbers with even denominator, $F$ is not differentiable at all the rational numbers with even denominator.
+
+Moral: There exists a continuous function on $[0,1]$ that is not differentiable at any rational number with even denominator. (Dense set)
+
+#### Weierstrass function
+
+$$
+g(x)=\sum_{n=0}^{\infty} a^n \cos(b^n \pi x)
+$$
+
+where $0<a<1$ and $ab>1+\frac{3}{2}\pi$.
+
+$g(x)$ is continuous on $\mathbb{R}$ but nowhere differentiable.
+
+_If we change our integral, will be differentiable at most points?_

pages/Math4121/Math4121_L14.md

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-# Lecture 14
+# Math 4121 Lecture 14
+
+## Recap
+
+### Hankel developedn Riemann's integrabilty criterion.
+
+#### Definition 
+
+Given an interval $I\subset[a,b]$, $f:[a,b]\to\mathbb{R}$ the oscillation of $f$ at $I$ is
+
+$$
+\omega(f,I) = \sup_I f - \inf_I f
+$$
+
+#### Theorem 
+
+A bounded function $f$ is Riemann integrable if and only if
+
+

pages/Math4121/Math4121_L2.md

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-# Lecture 2
+# Math4121 Lecture 2
 
 ## Chapter 5: Differentiation
 

pages/Math4121/Math4121_L3.md

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-# Lecture 3
+# Math4121 Lecture 3
 
 ## Continue on Differentiation
 

pages/Math4121/Math4121_L4.md

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-# Lecture 4
+# Math4121 Lecture 4
 
 ## Chapter 5. Differentiation
 

pages/Math4121/Math4121_L5.md

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-# Lecture 5
+# Math4121 Lecture 5
 
 ## Continue on differentiation
 

pages/Math4121/Math4121_L6.md

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-# Lecture 6
+# Math4121 Lecture 6
 
 ## Chapter 6: Riemann-Stieltjes Integral
 

pages/Math4121/Math4121_L7.md

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-# Lecture 7
+# Math4121 Lecture 7
 
 ## Continue on Chapter 6
 

pages/Math4121/Math4121_L8.md

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-# Lecture 8
+# Math4121 Lecture 8
 
 ## Continue on Riemann-Stieltjes Integral
 

pages/Math4121/Math4121_L9.md

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-# Lecture 9
+# Math 4121 Lecture 9
 
 Exam next week.