Kelvin Lee

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 continuity and uniform continuity ?
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 \text{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 $f(x)=g(x)$.
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 $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and that $f^{\prime}(x)$ exists and is bounded on $\mathbb{R}$. Show that $f$ is uniformly continuous on $\mathbb{R}$.
27. Let $f$ be a (Darboux) integrable function on $[a, b]$ and $F$ a differentiable function on $[a, b]$ with $F^{\prime}(x)=f(x)$ except for finitely many $x \in[a, b] .$ Show that $f^{\prime \prime}$ is integrable as well and conclude: $$ \int_{a}^{b} f=F(b)-F(a). $$ 28. Give a complete proof of the integral criterion for convergence of series. Namely for a monotone decreasing function $f:[0, \infty) \longrightarrow \mathbb{R}$ with $f(x) \geq 0$ for all $x \geq 0$ and $$ \lim _{b \rightarrow \infty} \int_{0}^{b} f<\infty $$ the series $\sum_{m=1}^{\infty} f(m)$ converges (absolutely).
29. If $f(x)$ and $g(x)$ are uniformly continuous on $\mathbb{R}$, then $f \cdot g$ is uniformly continuous on $\mathbb{R}$.
Answer : False.
30. $(10$ points) A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called periodic if there exists a $T>0$ such that $f(x)=f(x+T)$ for all $x \in \mathbb{R} .$ Suppose that $f$ is a periodic function that is differentiable on $\mathbb{R}$. Show that there exists $x \in \mathbb{R}$ such that $f^{\prime}(x)=0$.
31. Let $[a, b]$ be an interval and $c \in(a, b)$. Define a function $f$ on $[a, b]$ as follows:
$$ f(x)=\begin{cases} 0 & x \neq c
1 & x=c \end{cases}.$$ Show that $f$ is integrable on $[a, b]$, and find $\int_{a}^{b} f$.
32. Let $g$ be an integrable function on $[a, b]$ and suppose that $h(x)=g(x)$ for all but one $x$ in $[a, b] .$ Show that $h$ is integrable and that $\int_{a}^{b} g=\int_{a}^{b} h$.
33. Consider $f:[a, b] \rightarrow \mathbb{R}$. Suppose that $f^{\prime}$ is bounded where it exists (do not assume it exists everywhere). Then, $f$ is bounded.
34. 7. Let $f$ be an infinitely differentiable function on $\mathbb{R}$. Then the Taylor series of $f$ at any point $a \in \mathbb{R}$ converges to $f$ in some neighborhood of $a$.
35. Define $$ f_{n}(x)=\sum_{k=1}^{n} \frac{1}{k^{2} x^{2}} $$ defined on $(0, \infty)$. The $f_{n}$ converge uniformly.
36. Suppose $a_{n}>0$ and $\sum a_{n}$ converges. Then $\sum \frac{a_{n}+a_{n+1}}{2}$ converges.
37. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to satisfy the Lipshitz condition of order $\alpha$ at $a \in \mathbb{R}$ if there is a constant $M$ and a neighborhood of $a$ such that $$ \left|f(x)-f(a)\right|<M|x-a|^{\alpha} $$ Show that if $f$ has the Lipschitz condition of order $\alpha$ for $\alpha>0$, then $f$ is continuous. Give an example of a function $f$ which has Lipschitz condition of order 0 which is not continuous at $c$.
38. Show that the function $f(x)=x^{a}$ is uniformly continuous for $a>1$.
39. If $f: X \rightarrow Y$ is continuous and is not a constant function (i.e. $f(X)$ has more than one point), and $y \in Y$ is an isolated point, then $f^{-1}(\{y\})$ consists only of isolated points.
40. Let $E \subset \mathbb{R}$ be a closed subset. There is some $F \subset \mathbb{R}$ whose set of limit points is exactly $E$. (Note that isolated points of $E$ are not limit points of $E$.)
41. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. If a sequence $p_{n}$ in $\mathbb{R}$ diverges, then $f\left(p_{n}\right)$ also diverges.
42. Let $f$ be a function defined on $(0,1]$, and suppose $f$ is integrable on every interval $[c, 1]$ for $c \in(0,1)$. Define: $$ \int_{0}^{1} f d x=\lim _{c \rightarrow 0} \int_{c}^{1} f d x $$ Show that if $f$ is defined on $[0,1]$ and is integrable, then this definition agrees with the usual one. Find an example of a function $f:(0,1] \rightarrow \mathbb{R}$ such that the above integral exists for $f$ but not for $|f|$.
43. Suppose $f$ is bounded and real, and suppose $f^{2}$ is Riemann integrable (with respect to $\left.\alpha(x)=x\right)$. Does it follow that $f$ is Riemann-integrable?
44. Suppose $f$ is bounded and real, and suppose $f^{3}$ is Riemann integrable (with respect to $\left.\alpha(x)=x\right)$. Does it follow that $f$ is Riemann-integrable?
45.Suppose that $f$ is infinitely differentiable everywhere. Suppose that there is some $L>0$ such that $\left|f^{(n)}(x)\right|<L$ for all $n$ and $x \in \mathbb{R}.$ Further suppose that $f\left(\frac{1}{n}\right)=0$ for all $n \in \mathbb{N} .$ Show that $f(x)=0$ everywhere.
46. Define the integral: $$ \int_{a}^{\infty} f d x:=\lim _{b \rightarrow \infty} \int_{a}^{b} f d x $$ If it exists, we say it converges. Suppose that $f(x) \geqslant 0$ and $f$ decreases monotonically on $[1, \infty)$. Show that $$ \int_{1}^{\infty} f d x $$ converges if and only if $$ \sum_{n=1}^{\infty} f(n) $$ converges.
47. Suppose $\alpha$ is monotonically increasing and continuous on $[a, b]$ and $p \in[a, b] .$ Define $f$ by $f(p)=1$ and $f(x)=0$ for $x \neq p$. Show directly (do not cite a theorem other than the basic “Cauchy criterion” for integrability) that $f$ is Riemann integrable with respect to $\alpha$ and that $\int_{a}^{b} f d \alpha=0$.
48. Prove that $$ \sum_{n=2}^{\infty} \frac{1}{n(\log (n))^{p}} $$ converges for $p>1$ and diverges for $p \leqslant 1$.
49. Prove that if $a_{n}$ is a decreasing sequence of real numbers and if $\sum a_{n}$ converges, then $\lim n a_{n}=0$.
50. Let $f$ and $g$ be continuous functions on $[a, b]$ that are differentiable on $(a, b)$. Suppose that $f(a)=f(b)=0 .$ Prove that there exists $x \in(a, b)$ such that $g^{\prime}(x) f(x)+f^{\prime}(x)=0$.