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 ]$. 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 $f(x)=g(x)$.
15. Let $\left\{f_{n}\right\}$ be a sequence of continuous functions on $[a, b]$ that converges uniformly to $f$ on $[a, b] .$ Show that if $\left\{x_{n}\right\}$ is a sequence in $[a, b]$ and if $x_{n} \rightarrow x$, then $\lim _{n \rightarrow \infty} f_{n}\left(x_{n}\right)=f(x)$.
16. Find an example or prove that the following does not exist: a monotone sequence that has no limit in $\mathbb{R}$ but has a subsequence converging to a real number.
17. Consider a continuous function $f$ on $(0, \infty)$, and suppose that $f$ is a uniformly continuous on $(0, a)$ for all $a>0$. Then $f$ must be a uniformly continuous function on $(0, \infty)$.
18. Consider a sequence $\left(f_{n}\right)_{n=1}^{\infty}$ of continuous functions on $[0,1]$. Suppose that $\left(f_{n}\right)$ converges pointwise to a function $f$ on $[0,1]$, and that $$ \lim _{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) d x=\int_{0}^{1} f(x) d x $$ Then, $\left(f_{n}\right)$ must converge to $f$ uniformly on $[0,1]$.
19. Suppose that a sequence of functions $\left(f_{n}\right)_{n=1}^{\infty}$ converges to $f$ uniformly on $(0,1)$. Then, the sequence $\left(f_{n}^{3}\right)_{n=1}^{\infty}$ converges to $f^{3}$ uniformly on $(0,1)$.
20. Let $0<a<1$ be a fixed number. Suppose that a sequence of functions $\left(g_{n}\right)_{n=1}^{\infty}$ on $[0, a]$ satisfies $\left|g_{n}(x)\right| \leq x^{n}$ for all $x \in[0, a]$ and for all $n \in \mathbb{N}$. Then, $\sum g_{n}(x)$ is a uniformly convergent infinite series.
21. Suppose that a sequence of functions $\left(f_{n}\right)_{n=1}^{\infty}$ on $[0,1]$ converges uniformly to $f$ on $[0,1]$. Let $g$ be a continuous function on $[0,1]$. Prove that $\left(f_{n} g\right)_{n=1}^{\infty}$ converges uniformly to $f g$ on $[0,1]$.
22. $\left(10\right.$ points) Let $\sum a_{n}$ be a convergent series and $\left(f_{n}\right)$ be a sequence of real-valued functions defined on $S \subset \mathbb{R}$ such that $$ \left|f_{n+1}(x)-f_{n}(x)\right|<a_{n}, \quad \forall n \in \mathbb{N}, \forall x \in S $$ Prove that $\left(f_{n}\right)$ is uniformly Cauchy on $S$ and hence it is uniformly convergent on $S$.
23. Let $\alpha$ be a bounded, monotonically increasing function on $\mathbb{R}$. What is $$ \int_{-\infty}^{\infty} 1 d \alpha ? $$ 24. Suppose $f$ is continuous on $[a, b]$ and $\alpha$ is continuous and strictly increasing. Show that if $$ \int_{a}^{b} f^{2}(x) d \alpha=0 $$ then $f$ is identically 0 on $[a, b]$.
25. Suppose that $f$ and $g$ are continuous functions on $[a, b]$ such that $\int_{a}^{b} f=\int_{a}^{b} g .$ Prove that there exists $x \in[a, b]$ such that $f(x)=g(x)$.
26. Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is a continuous function and that $f^{\prime}(x)$ exists and is bounded on $\mathbb{R}$. Show that $f$ is uniformly continuous on $\mathbb{R}$.