math104-s21:hw9
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| 2. (2 points) Prove that if $f$ is Lipschitz continuous then $f$ is uniformly continuous. | 2. (2 points) Prove that if $f$ is Lipschitz continuous then $f$ is uniformly continuous. | ||
| - | 3. (4 points) Let $f_n: X \to \R$ be a sequence of functions on a metric space $X$. Suppose $f_n$ converges to $f$ pointwise, and $f_n$ are Lipschitz continuous with a common $K$ as a Lipschitz constant, namely, for any $x, y \in X$, and any $n \in \N$, we have $ |f_n(x) - f_n(y) | \leq K d(x,y). $ Is it true that $f_n$ converges to $f$ uniformly? If true, prove it; otherwise find counterexample. | + | 3. (2 points) Let $f_n: X \to \R$ be a sequence of functions on a metric space $X$. Suppose $f_n$ converges to $f$ pointwise, and $f_n$ are Lipschitz continuous with a common $K$ as a Lipschitz constant, namely, for any $x, y \in X$, and any $n \in \N$, we have $ |f_n(x) - f_n(y) | \leq K d(x,y). $ Is it true that $f_n$ converges to $f$ uniformly? If true, prove it; otherwise find counterexample. |
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math104-s21/hw9.1616106755.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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