I have kept most of my notes handwritten throughout the semester. Instead of taking tons of photos of my handwritten notes, I will pick some of the important concepts and theorems to put on this website so that I can review the final in a more efficient way. Feel free to comment on my website:)
1. Number System:
Natural numbers:
- allowed operations: +, *
- exists successor
- 1 belongs to N, n belongs to N, n+1 belongs to N
- basis of induction
Integers:
- distinguished 0
Rationals:
- example of field
- ordered field
Completeness axiom:
- If S is bounded above, then sup(S) exists. If also bounded below, inf(S) = -sup(-S)
Archimedian property:
- if a,b>0, then there exists a natural number n such that n*a>b. Prove by contradiction
Rational roots theorem: - If r = c/d != 0 a rational number, and r satisfies cn*x^n + cn-s*x^n-1 +…+c0=0 with ci in Z, cn, c0 not equal to 0, then d|Cn, and c|C0
2. Sequences:
Sequence and limit:
- use epsilon delta proof
Note: all convergent sequences are bounded.
Useful results of limits: preserves addition, subtraction, multiplication, and division when denominator not equal to 0.
Note:
- let An be bounded, then An converges iff limsup(an) = liminf(an)
- for all epsilon, there is a natural number N>0 such that for all n>N, an<limsup(an)+epsilon
Bounded monotone sequence: - Bounded monotone sequences converge.
Subsequence:
Thm: let Sn be s sequence, and a real number t, then (Sn) has a subsequent converges to t iff for all epsilon>0, the set Aepsilon{natural number n| abs(Sn-t) < epsilon} is infinite.
Note:
- Every sequence has a monotone subsequence.
- Therefore, every bounded sequence has a convergent subsequence(bolzano-welstress)
Subsequential limit:
- The limsup(Sn) and liminf(Sn) are subsequential limits of Sn. And sup(S*) = limsup(Sn), inf(S*) = liminf(Sn) where S* is the set of all subsequential limits of Sn. If S* has a single element, then all subsequent converge to lim(Sn)
- The S* for bounded Sn is closed.
Important inequality given nonzero real sequence an:
$\liminf |a_{n+1}/a_n| \leq \liminf |a_n|^\frac{1}{n} \leq \limsup |a_n|^\frac{1}{n} \leq \limsup |a_{n+1}/a_n|$
Technique:
- When proving Sn diverges to infinity, try to find for all M>0 there exists N>0 such that for all n>N, Sn>M
- Triangle inequality
3. Metric Space
Properties of distance metrics:
- d(x,y) >= 0 and only equal to zero when x=y
- d(x,y) = d(y, x)
- d(x,y) + d(y,z) >= d(x,z)
Note:
Convergence of Sn implies cauchy, but the converse is true only when the metric space is complete.
- A metric space is complete if every cauchy sequence in it has a limit.
Open and close:
- finite intersection of open set is open
- arbitrary union of open set is open
- finite union of closed set is closed
- arbitrary intersection of closed set is closed
Closure property:
- If x in the closure of the set S, then for all epsilon>0, epsilon ball(x) intersects with E is not empty.
Heine-Borel:
- Let K be a subset of Rn, then K compact iff K is closed and bounded.
note(converse is true only in Rn)
Sequential compactness:
- S is sequential compact if for all infinite sequence Sn in S has a convergent subsequence. Note: Sequential compact ⇔ covering compact
Interesting observations:
- the boundary of a ball is not always a sphere (consider a discrete metric).
- closure of Q is R
- with L1 metric, an open ball is a diamond
- with L infinity metric, an open ball is a square
4. Series:
Note:
- The cauchy of a series implies it converges.
Useful formula:
- 1+x+x^2+…+x^n = (1-x^(n+1))/(1-x)
- infinity summation of a*r^n starting from index 0 = a/(1-r) where |r| < 1
Series tests:
- comparison test
- root test - absolute convergence
- ratio test - absolute convergence
- alternating series test
- integral test (continuous, positive, decreasing)
5. Functions between metric spaces:
Reminder:
- Notice the difference between uniform convergence and convergence on a single point.
Different definitions of continuity:
- A function F from X to Y is continuous iff for all V in Y open, F^-1(V) is open.
- A function F from X to Y is continuous iff for all p in X' we have f(p) = lim f(x):x →p
Related theorem:
- let F:X→Rn, with F(x) = (f1(x), f2(x),…,fn(x)) then f is continuous ⇔ fi:x→R are continuous for all
i = 1,…,n
induced topology:
- let X be a topological space, S is a subset of X, we can equip S with induced topology E within S is open in S iff there exists open subset E2 such that E = intersection of S and E2.
Thm:
- f:X → Y continuous, then if E within X, then f(E) is compact in Y
- f:X → Y continuous, and X is compact, then f is uniformly continuous.
- f:X → Y continuous, and X is compact, and f is a bijection, then f inverse if continuous.
Connectedness:
Let X be a topological space:
- X is connected iff the only subset of X that is both open and closed are X and empty set.
- X is not connected iff there exists U, V in X nonempty, open, and disjoint such that X = U union V.
Thm:
- f:X → Y continuous, and X is connected, then f(X) is connected.
- IVT
- MVT
Note:
F is continuous at p iff f(p) = f(p+) = f(p-)
- if f(p+) and f(p-) exist, but not continuous at p, then first kind discontiuity
- otherwise, second kind discontinuity.
Thm:
- if f is monotone increasing on (a,b), then f(x+) and f(x-) exists at every point in (a,b) and has at most countably many discontinuities which are all first kind.
Convergence of functions:
- pointwise convergence vs. uniform convergence
- bump function as example
- pointwise convergence does not preserve integration nor differentiation
Uniform convergence:
- l infinity metric: d(fn, f) = 0 as n goes to infinity.
- uniform cauchy implies uniform convergence
- preserves continuity
Thm:
Let K be compact metric space, fn:K→R
1. fn is continuous for all n;
2. fn(x)→f(x) and f is continuous for all x
3. fn(x)>=fn+1(x) for all x and n
then fn converges to f uniformlly.
6. Differentiability and Integrability
Prop:
- Differentiability implies continuity
- Differentiability ⇒ there exists u(x) such that f(x) = f(p)+(x-p)f'(p) + (x-p)u(x) where lim u(x) as x goes to p = 0
Caveat:
- f' exists everywhere doesn't imply that f' is continuous.
Rolle's theorem
- if f continuous on [a,b] and f' exists for all x in (a,b) if f(a) = f(b) then there exists c in (a,b) such that f'© = 0
Generalized Mean Value Theorem
- if f, g continuous and differentiable on (a,b), then there exists c in (1,b) such that [f(b) - f(a)]g'© = [g(b) - g(a)]f'\©
Corollary:
- if f differentiable and bounded by M>0, then f is uniformally continuous.
L'Hopital Rule
- lim f(x)/g(x) = lim f'(x)/g'(x) if lim f(x)/g(x) = 0/0 or inf/inf
Taylor's Theorem
- given f that is smooth till n-1 degree, and is continuous on on interval $[a, b]$, and given $\alpha, \beta \in [a, b]$, we have $$f(\beta) = \sum_{i=0}^{n-1}\frac{f^{(i)}(\alpha) (\beta - \alpha)^i}{i !} + \frac{f^{(n)}(\gamma)(\beta - \alpha)^n}{n!}$$ for some $\gamma$ between $\alpha$ and $\beta$.
Riemman Integral
Idea:
- A partition P of interval [a,b] is a = x0<x1<…xn=b.
- Define U(P,f) and L(P,f)
- Let U(f) = inf U(P,f) and L(f) = sup L(P,f)
- f is integrabvle if U(f) = L(f)
Generalization:
- Let alpha be a monotone increasing function, define interval delta alpha = alpha(xi) - alpha(xi-1)
- If U(P, alpha) = L(P, alpha), then it is riemman-stieltjes integrable with respect to alpha.
Refinement lemma:
- If Q is a refinement of P on [a,b],then the approximated integral bounds get better: Lp ⇐ Lq ⇐ Uq ⇐ Up
Cauchy condition of integral:
- f is integrable w.r.t. alpha iff for all epsilon > 0, there exists p partition such that Up - Lp < epsilon.
Related theorems:
- If f is continuous on [a,b], then f is integrable w.r.t. alpha on [a,b]
- If f is monotonic, and alpha continuous, then f is integrable
- If f is discontinuous only at finitely many points and alpha is continuous where f is discontinuous, then f is integrable
- if f:[a,b]→[m,M] and phi:[m,M]→R is continuous, if f is integrable wrt alpha, then h = phi composed with f is integrable wrt alpha
- integration operation is linear in both f and alpha
- composition of integrable functions is integrable
- |f| is integrable if f is integrable
- let alpha be monotone increasing and alpha' exists and is riemman integrable wrt to dx, then bounded f integrable ⇔ f*alpha'is integrable
- change of variable
- integration by parts
- uniform convergence preserves integration
Questions (updating):
1. Are sup and inf guaranteed to exist? If sup = infinity, do we say sup exists or not?
2. If Sn is bounded, is it guaranteed that limsup and liminf always exist?
3. Does pointwise convergence preserves continuity like uniform convergence does?
4. Extra question from hw:Let $f(x)$ be a differentiable function on $[-1,1]$ with $f'(x)$ continuous. Assume $x=0$ is the unique global minimum of $f$, i.e., for any $x \neq 0$, we have $f(x) > f(0)$. Is it true that there exists a $\delta > 0$, such that $f'(x) < 0$ for $x \in (-\delta, 0)$ and $f'(x)>0$ for $x \in (0 ,\delta)$? (Just as the case if $f(x)=x^2$) –I think should be true?
* I’m having a hard time to imagine why why would 1/x be integrable on R? Since it is discontinuous only at x=0, and f(x) = x is continuous at x=0, it is integrable. But what about the strange behavior near x=0?
5. In hw11, problem 1, the reverse direction, h ⇐ f'(y) was left as an exercise. However I'm having trouble to prove this. Any hint?
6. Givan P1 = {1,4,6}, P2 = {1, 3, 3.5, 5, 5.6, 6}, is P2 a refinement of P1?
7. What is the relationship of boundedness to integrability? Does one imply another? aka not bounded ⇒ not integrable?
8. Let a1 be any real number satisfying 0 < a1 < 1 , and define a2, a3, . . . recursively via an+1 := cos(an). Why would this sequence be monotone? I plugged a0 = 0.5 into my calculator and it is clearly not monotone. If so, how should we prove that this sequence converges?
9. What is an efficient way, if any, to test uniform convergence and uniform continuity?
10. Is it true that if F is differentiable then it is integrable and vice versa?
11. What does the notation: Sum{from n=1 to infinity}fn converges uniformly mean?
12. Determine whether sum{n=1,inf} 1/(n+0.5) converges or diverges?
13: Let f : [a, b] → R be a Riemann integrable function. Let g : [a, b] → R be a
function such that {x ∈ [a, b]; f(x) = g(x)} is finite. Show that g(x) is Riemann integrable and the integral of f(x) on [a,b] is equal to that of g(x). Does the conclusion still hold when {x ∈ [a, b]; f(x) = g(x)} is countable?
14: When do we have equality in the inequality |integral f| < = integral |f| ?
15: What are ways of proving connectedness?
16: What is the epsilon room proof?
17: What are good strategies to coming up with the right partitions for integration proofs?
18: Why is change of variables useful? And what is its general form?
19: What is the Second Mean Value Theorem for Integrals?
20: Is the boundary of an open ball always a sphere?
21: What is the difference between discrete metric and continuous metric?
22: What does the iota function mean?
23: How can we apply change of variables to usub?
24: In usub, I don't remember that u needs to be strictly increasing. But in the requirement of change of variables, phi needs to be strictly increasing. How should we explain this?
25: When can we not use Lopital's Rule?
26: Is the set [0,1]∩Q compact or not? (from mt2)
27: For the radius of convergence for power series, we used a similar approach of root test. Is there a ratio test analog of this proof?
28: Is it true that uniform convergence preserves integration in general? What about pointwise convergence?
29: Is it true that uniform convergence preserves derivatives in general? What about pointwise convergence?
30: What does the alpha function mean in the context of integration?
31: How to test if taylor's expansion would fail or not?
32: What does the step function mean in the context of integration? What about the infinite sum of step function?
33: How to do decimal digit expansion?
34: Why is Archimedian property useful?
35: What is exactly a topology? Is it just an abstract space consist of open and closed sets?
36: How to show a space is complete?
37: What are the implications of dense subsets?
38: Let f : [a, b] → R be a continuous function. Show that there exists c ∈ (a, b)
such that integral from a to b of f(x)dx = f©.
39: Discuss the convergence or divergence of the Bertrand Integrals (problem 7.21 from problem book.
40: What are some techniques of picking epislon and delta? Is this mostly a reverse-engineering process?
41: What are the necessary conditions for a function to be integrable? and differentiable?
42: What are efficient ways to prove discontinuity?
43: Is the criterion for induced metric always valid? When might induced metric not be useful?
44: Prove that if the function f : I → R has a bounded derivative on I, then f is
uniformly continuous on I. Is the converse true? What is the relationship between differentiability and uniform continuity?
45: Show that the equation ex = 1 − x has one solution in R. Find this solution.
46: Let f : [0,∞) → R differentiable everywhere. Assume that limx→∞ f(x)+f(x) = 0.Show that limx→∞ f(x) = 0.
47: Prove that if c is an isolated point in D, then f is automatically continuous at c.
48: Let f(x) = x sin(1/x) for x = 0 and f(0) = 0. Prove that f is continuous at
x = 0
49: Find an example of a function that is discontinuous at every real number.
50: Find an example of a function f discontinuous on Q and another function g
discontinuous at only one point, but g ◦ f is nowhere continuous.
51: Let f(x)=[x] be the greatest integer less than or equal to x and let g(x) = x−[x].
Sketch the graphs of f and g. Determine the points at which f and g are continuous.