math104-s21:s:victoriatuck
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math104-s21:s:victoriatuck [2021/05/11 23:27] 73.158.208.111 [Questions] |
math104-s21:s:victoriatuck [2026/02/21 14:41] (current) |
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| a) Question: What is a counterexample to "If $A$ is closed, then $f(A)$ is closed" | a) Question: What is a counterexample to "If $A$ is closed, then $f(A)$ is closed" | ||
| - | Answer: Let $f := arctan(x)$. Then. $(-\infty, \infty) -> (-\frac{\pi}{2}, | + | Answer: Let $f := arctan(x)$. Then $(-\infty, \infty) -> (-\frac{\pi}{2}, |
| b) Question: What is a counterexample to "If $B$ is connected, then $f^{-1}(B)$ is connected."? | b) Question: What is a counterexample to "If $B$ is connected, then $f^{-1}(B)$ is connected."? | ||
| - | Answer: | + | Answer: |
| 6. Question: Given an open cover $\mathcal{U}$ of $f(E)$ where f is a continuous function and E is a compact set, why is $\{f^{-1}(U) : U \in \mathcal{U}\}$ a cover of E? [Question from Ross Theorem 21.4] | 6. Question: Given an open cover $\mathcal{U}$ of $f(E)$ where f is a continuous function and E is a compact set, why is $\{f^{-1}(U) : U \in \mathcal{U}\}$ a cover of E? [Question from Ross Theorem 21.4] | ||
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| Answer: In Ross 168, we define the inverse $f^{-1}(U)$ to include any value $s$ st $f(s) \in U$. So $f^{-1}([1, inf)) = (-inf, -1] \cup [1, inf)$. | Answer: In Ross 168, we define the inverse $f^{-1}(U)$ to include any value $s$ st $f(s) \in U$. So $f^{-1}([1, inf)) = (-inf, -1] \cup [1, inf)$. | ||
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| + | 8. Question: | ||
math104-s21/s/victoriatuck.1620775625.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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