math104-s21:s:dingchengyang
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math104-s21:s:dingchengyang [2021/05/09 21:23] 73.132.71.37 |
math104-s21:s:dingchengyang [2026/02/21 14:41] (current) |
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| 13: Let f : [a, b] → R be a Riemann integrable function. Let g : [a, b] → R be a | 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? | 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' | ||
| + | 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' | ||
| + | 32: What does the step function mean in the context of integration? | ||
| + | 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(c).\\ | ||
| + | 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.\\ | ||
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math104-s21/s/dingchengyang.1620595418.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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