Transcription of Section 18. Continuous Functions
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Section 18. Continuous Functions
18. Continuous Functions1 Section isthefundamental concept in topology! When you hear that a coffee cup and a doughnut are topologically equivalent, this is really a claimabout the existence of a certain Continuous function (this idea is explored in depthin Chapter 12, Classification of Surfaces ). We start by reviewing some continuityideas from Analysis 1 (MATH 4217/5217). standard definition of continuity of a real valued function of a realvariable at a pointx0in the domain of the functionf,D(f), is as follows ( , page 2) a function andx0 D(f). Thenfiscontin-uous at pointx0iffor all >0 there exits ( )>0 such thatfor all|x x0|< ( ) andx D(f) we have|f(x) f(x0)|< .We then say thatfis Continuous a setA Riffis Continuous at each point following is a consequence of the previous definition (see Theorem 4-5in the Analysis 1 notes mentioned above):Theorem :X Y.
Jun 11, 2016 · Note. As you can see in the proof of the previous theorem, we can replace “closed” with “open” to get the following. Corollary 18.A. The Pasting Lemma for Open Sets. Let X = A∪ B where A and B are open in X. Let f : A → Y and g : B → Y be continuous. If f(x) = g(x) for all x ∈ A ∪ B, then f and g combine (or “paste”)
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