notes / Math 55b / chapter 1: Metric and Topological Spaces / From my notebook
N2 Continuity
Definition N2.1 (Continuity). Given two metric spaces and , a map (defined everywhere) is continuous if for every and every there is such that
Equivalently,
Theorem N2.2. A function is continuous if and only if for every open subset of , is open.
Proof. Assume is continuous, and let be open. Pick some point . Then , which is open, so there is an open ball, , such that . Since is continuous, there is such that
therefore . So is open.
Now assume that open implies is open in . Pick ,. Let , open in . So is open, therefore is open in , and it contains . Hence there is such that , so
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