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Advanced Calculus: Difference between revisions
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==Sequences== | ==Sequences== | ||
==Topology== | |||
===Closed=== | |||
The folllowing definitions of Closed Sets are equivalent. | |||
* (Order) | |||
* (Sequences) A set <math>S</math> is clsoed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>. | |||
* (Topology) | |||
Union of infinitely many closed sets can be open. | |||
Intersection of infinitely many open sets can be closed. | |||
===Compact=== | |||
===Metric Space=== | |||
==Continuity== | ==Continuity== | ||
===Definitions of Continuity=== | ===Definitions of Continuity=== | ||
The following | The following definitions of Continuity are equivalent. | ||
* (Order) A function <math>f</math> is continuous at <math>x_0</math> if for all <math>\epsilon</math> there exists <math>\delta</math> such that <math>|x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon</math> | * (Order) A function <math>f</math> is continuous at <math>x_0</math> if for all <math>\epsilon</math> there exists <math>\delta</math> such that <math>|x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon</math> | ||
* (Sequences) A function <math>f</math> is continuous at <math>x_0</math> if <math>\{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)</math> | * (Sequences) A function <math>f</math> is continuous at <math>x_0</math> if <math>\{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)</math> | ||