Advanced Calculus: Difference between revisions
(Created page with "Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book]. ==Sequences== ==Continuity== ===Definitions of Continuity=== The following 3 defi...") |
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* (Topology) A function <math>f</math> is continuous at <math>x_0</math> if for all open sets <math>V</math> s.t. <math>f(x_0) \in V</math>, <math>f^{-1}(V)</math> is an open set. | * (Topology) A function <math>f</math> is continuous at <math>x_0</math> if for all open sets <math>V</math> s.t. <math>f(x_0) \in V</math>, <math>f^{-1}(V)</math> is an open set. | ||
** The preimage of an open set is open. | ** The preimage of an open set is open. | ||
** | ** Continuous functions map compact sets to compact sets. | ||
==Differentiation== | ==Differentiation== | ||
Revision as of 04:08, 5 November 2019
Advanced Calculus as taught in Fitzpatrick's book.
Sequences
Continuity
Definitions of Continuity
The following 3 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.