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math104-s21:hw7 [2021/03/05 23:48] pzhou |
math104-s21:hw7 [2022/01/11 18:31] (current) 24.253.46.239 ↷ Links adapted because of a move operation |
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====== HW 7 ====== | ====== HW 7 ====== | ||
- | This week we discussed continuity of maps, the three equivalent definitions of continuity. Then on Tuesday, | + | On Tuesday, |
$\gdef\In\subset$ | $\gdef\In\subset$ | ||
- | Warm up: (no submission required) | + | ===== Warm up: ===== |
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+ | no submission required | ||
1. Let $f: X \to Y$ be a map between metric spaces. Check if the following statements are true or not. If you think the statement is false, give some counter example. If you think the statement is true, give some reasoning. | 1. Let $f: X \to Y$ be a map between metric spaces. Check if the following statements are true or not. If you think the statement is false, give some counter example. If you think the statement is true, give some reasoning. | ||
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3. Let $f: (0, \infty) \to \R$ be a map given by $f(x) = \sin(1/x)$, prove that $f$ is continuous. You may use that $\sin(x)$ is a continuous function. | 3. Let $f: (0, \infty) \to \R$ be a map given by $f(x) = \sin(1/x)$, prove that $f$ is continuous. You may use that $\sin(x)$ is a continuous function. | ||
- | Homework: | + | ===== Homework: |
- | {{ :math104: | + | {{ math104-s21: |
+ | Note that for problem 18, you only need to prove that $f$ is continuous at every irrational point (since we haven' | ||
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