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Section 18. Continuous Functions - East Tennessee State ...
faculty.etsu.eduJun 11, 2016 · 18. Continuous Functions 6 Theorem 18.2. Rules for Constructing Continuous Functions. Let X, Y, and Z be topological spaces. (a) (Constant Function) If f : X → Y maps all of X into a single point y0 ∈ Y, then f is continuous. (b) (Inclusion) if A is a subspace of X, the inclusion function j : A → X is continuous.
Continuous Functions on Metric Spaces
www.math.ucdavis.eduspace B(X;Y), so it is a complete metric space. 4 Continuous functions on compact sets De nition 20. A function f : X !Y is uniformly continuous if for ev-ery >0 there exists >0 such that if x;y2X and d(x;y) < , then d(f(x);f(y)) < . Theorem 21. A continuous function on a compact metric space is bounded and uniformly continuous. Proof.