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--- math104-s21:s:dingchengyang [2021/05/06 23:54]
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+++ math104-s21:s:dingchengyang [2026/02/21 14:41] (current)
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 i = 1,...,n\\
-** induced topology:**\\
+** 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:
+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.\\
@@ Line 149: / Line 149: @@
 - otherwise, second kind discontinuity.\\
-Thm:
+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.\\
@@ Line 169: / Line 169: @@
 then fn converges to f uniformlly.\\
-** 5. Differentiability and Integrability**
+** 6. Differentiability and Integrability**\\
-Prop:
+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\\
@@ Line 177: / Line 177: @@
 - f' exists everywhere doesn't imply that f' is continuous.\\
-** Rolle's theorem**
+** 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'(c) = 0\\
-** Generalized Mean Value Theorem**
+** 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'(c) = [g(b) - g(a)]f'(c)\\
+- if f, g continuous and differentiable on (a,b), then there exists c in (1,b) such that [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'\(c)\\
 Corollary:\\
 - if f differentiable and bounded by M>0, then f is uniformally continuous.\\
-** L'Hopital Rule**
+** 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**
+** 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):\\
-* Are sup and inf guaranteed to exist? If sup = infinity, do we say sup exists or not? \\
+. Are sup and inf guaranteed to exist? If sup = infinity, do we say sup exists or not? \\
-* If Sn is bounded, is it guaranteed that limsup and liminf always exist? \\
+. If Sn is bounded, is it guaranteed that limsup and liminf always exist? \\
-* Does pointwise convergence preserves continuity like uniform convergence does? \\
+. Does pointwise convergence preserves continuity like uniform convergence does? \\
+. 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?\\
+. 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?\\
+. Givan P1 = {1,4,6}, P2 = {1, 3, 3.5, 5, 5.6, 6}, is P2 a refinement of P1?\\
+. What is the relationship of boundedness to integrability? Does one imply another? aka not bounded => not integrable?\\
+. 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?\\
+. What is an efficient way, if any, to test uniform convergence and uniform continuity?\\
+. Is it true that if F is differentiable then it is integrable and vice versa?\\
+. What does the notation: Sum{from n=1 to infinity}fn converges uniformly mean?\\
+. Determine whether sum{n=1,inf} 1/(n+0.5) converges or diverges?\\
+: 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?\\
+: When do we have equality in the inequality |integral f| < = integral |f| ?\\
+: What are ways of proving connectedness?\\
+: What is the epsilon room proof?\\
+: What are good strategies to coming up with the right partitions for integration proofs?\\
+: Why is change of variables useful? And what is its general form?\\
+: What is the Second Mean Value Theorem for Integrals?\\
+: Is the boundary of an open ball always a sphere?\\
+: What is the difference between discrete metric and continuous metric?\\
+: What does the iota function mean?\\
+: How can we apply change of variables to usub?\\
+: 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?\\
+: When can we not use Lopital's Rule?\\
+: Is the set [0,1]∩Q compact or not? (from mt2)\\
+: 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?\\
+: Is it true that uniform convergence preserves integration in general? What about pointwise convergence?\\
+: Is it true that uniform convergence preserves derivatives in general? What about pointwise convergence?\\
+: What does the alpha function mean in the context of integration?\\
+: How to test if taylor's expansion would fail or not?\\
+: What does the step function mean in the context of integration? What about the infinite sum of step function?\\
+: How to do decimal digit expansion?\\
+: Why is Archimedian property useful?\\
+: What is exactly a topology? Is it just an abstract space consist of open and closed sets?\\
+: How to show a space is complete?\\
+: What are the implications of dense subsets?\\
+: 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(c).\\
+: Discuss the convergence or divergence of the Bertrand Integrals (problem 7.21 from problem book.\\
+: What are some techniques of picking epislon and delta? Is this mostly a reverse-engineering process?\\
+: What are the necessary conditions for a function to be integrable? and differentiable?\\
+: What are efficient ways to prove discontinuity?\\
+: Is the criterion for induced metric always valid? When might induced metric not be useful?\\
+: 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?\\
+: Show that the equation ex = 1 − x has one solution in R. Find this solution.\\
+: Let f : [0,∞) → R differentiable everywhere. Assume that limx→∞ f(x)+f(x) = 0.Show that limx→∞ f(x) = 0.\\
+: Prove that if c is an isolated point in D, then f is automatically continuous at c.\\
+: Let f(x) = x sin(1/x) for x = 0 and f(0) = 0. Prove that f is continuous at\\
+x = 0\\
+: Find an example of a function that is discontinuous at every real number.\\
+: 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.\\
+: 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.\\

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