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Advanced Calculus: Difference between revisions
→Definitions of Continuity
(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== | ||