Here is the link to my personal course notes for this class. Notes (They might contain typos or logical errors.) They are created based on Ross's, Rudin's textbooks and Professor Zhou's lectures.

1. What's the difference between ?
2. What's the difference between pointwise convergence and uniform convergence?
3. Is a series of continuous functions necessarily continuous?
Answer: No, consider $f_n(x)=x^{n}-x^{n-1}$ and $\sum_n f_n(x)$ is not continuous.
4. (Yuwei Fan's practice final) Let $f:[a, b] \rightarrow \mathbb{R}$ be an integrable function. Prove that $$ \lim _{n \rightarrow \infty} \int_{a}^{b} f(x) \sin (n x) d x=0. $$ 5. (Yuwei Fan's practice final) Suppose that $f:[1, \infty) \rightarrow \mathbb{R}$ is uniformly continuous on $[1,\infty]$. Prove that there exists $M>0$ such that $\frac{|f(x)|}{x} \leq M$ holds for any $x \geq 1$.
6. What are some nice properties that continuity preserves?
Answer: Compactness, connectedness. 7. What does it mean intuitively for a set to be both closed and open?
8. What is the motivation behind the concept of compactness?
9. If $\{f_n\}$ are continuous, does it mean that its limit is also continuous?
10. (Yuwei Fan's practice final) For a bounded function $f:[0,1] \rightarrow \mathbb{R}$, define $$ R_{n}:=\frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n}\right) $$ (a) Prove that if $f$ is integrable, then $\lim _{n \rightarrow \infty} R_{n}=\int_{0}^{1} f(x) d x$.
(b) Find an example of $f$ that is not integrable, but $\lim _{n \rightarrow \infty} R_{n}$ exists.
11. If $f_{n} \rightarrow f$ uniformly on $S$, then $f_{n}^{\prime} \rightarrow f^{\prime}$ uniformly on $S$.
Answer : False.
12. If $f$ is differentiable on $[a, b]$ then it is integrable on $[a, b]$.
Answer : True.
13. If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on $[a, b]$, there is a sequence of polynomials whose uniform limit on $[a, b]$ is $f .$
Answer : True.
14. Let $f$ and $g$ be continuous functions on $[a, b]$ such that $\int_{a}^{b} f=\int_{a}^{b} g$. Show that there is an $x \in[a, b]$ such that $f(x)=g(x)$.