math104-f21:hw11
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math104-f21:hw11 [2021/11/05 23:05] pzhou created |
math104-f21:hw11 [2021/11/08 07:35] pzhou |
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| For the following question: if true, prove it; if false, give a counter-example. | For the following question: if true, prove it; if false, give a counter-example. | ||
| - | 1. Assume $f: X \to Y$ is uniformly continuous, and $g: Y \to Z$ is uniformly continuous. Is it true that $f \circ g: X \to Z$ is uniformly continuous? | + | 1. Assume $f: X \to Y$ is uniformly continuous, and $g: Y \to Z$ is uniformly continuous. Is it true that $g \circ f: X \to Z$ is uniformly continuous? |
| 2. If $f: (0,1) \to (0,1)$ is a continuous map, is it true that there exists a $x \in (0,1)$, such that $f(x) = x$? What if one change the setting to $f: [0,1] \to [0,1]$ being continuous, does your conclusion change? | 2. If $f: (0,1) \to (0,1)$ is a continuous map, is it true that there exists a $x \in (0,1)$, such that $f(x) = x$? What if one change the setting to $f: [0,1] \to [0,1]$ being continuous, does your conclusion change? | ||
math104-f21/hw11.txt · Last modified: 2022/01/11 08:36 by pzhou
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