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第九周。

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2023-04-22 11:59:26 +08:00
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13多变量函数的连续性.tex
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@@ -477,7 +477,7 @@
\section{连续函数/映射的性质}
\begin{definition}[向量值函数的一致连续]
对$D \subset \ndreal$，$\boldf: D \to \realnum^m$。如果对任意的$\varepsilon > 0$，存在$\delta$使得对任意的$\bvec{x}^\prime, \bvec{x}^{\prime \prime} \in D$只要满足$\norm{\bvec{x}^\prime - \bvec{x}^{\prime \prime} < \delta}$都有$\norm{\boldf(\bvec{x}^\prime) - \boldf(\bvec{x}^{\prime \prime})} < \varepsilon$，那么称映射\boldf 在$D$上一致连续。
对$D \subset \ndreal$，$\boldf: D \to \realnum^m$。如果对任意的$\varepsilon > 0$，存在$\delta$使得对任意的$\bvec{x}^\prime, \bvec{x}^{\prime \prime} \in D$只要满足$\norm{\bvec{x}^\prime - \bvec{x}^{\prime \prime}} < \delta$都有$\norm{\boldf(\bvec{x}^\prime) - \boldf(\bvec{x}^{\prime \prime})} < \varepsilon$，那么称映射\boldf 在$D$上一致连续。
\end{definition}
\begin{theorem}[紧致集上的连续性]