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> | ||
Revision as of 04:14, 5 November 2019
Advanced Calculus as taught in Fitzpatrick's book.
Sequences
Topology
Closed
The folllowing definitions of Closed Sets are equivalent.
- (Order)
- (Sequences) A set \(\displaystyle S\) is clsoed if it contains all its limit points. That is \(\displaystyle \forall \{x_i\} \subseteq S\), \(\displaystyle \{x_i\} \rightarrow x_0 \implies x_0 \in S\).
- (Topology)
Union of infinitely many closed sets can be open.
Intersection of infinitely many open sets can be closed.
Compact
Metric Space
Continuity
Definitions of Continuity
The following definitions of Continuity are equivalent.
- (Order) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all \(\displaystyle \epsilon\) there exists \(\displaystyle \delta\) such that \(\displaystyle |x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon\)
- (Sequences) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if \(\displaystyle \{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)\)
- (Topology) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all open sets \(\displaystyle V\) s.t. \(\displaystyle f(x_0) \in V\), \(\displaystyle f^{-1}(V)\) is an open set.
- The preimage of an open set is open.
- Continuous functions map compact sets to compact sets.