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Theorem 7.10 (Pasting lemma). Let X = A U B, where A, B are closed in X. Let f : A → Y and g : B → Y be continuous functions that agree on A ŋ B. Then the function h : AUB → Y defined by h = ƒ on A and h = g on B is continuous.

Theorem 7.10 (Pasting lemma). Let X = A U B, where A, B are closed in X. Let f : A → Y and g : B → Y be continuous functions that agree on A ŋ B. Then the function h : AUB → Y defined by h = ƒ on A and h = g on B is continuous.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Could you explain how to show 7.10 in detail?

Definition. Let X and Y be topological spaces. A function or map f : X → Y is a continuous function or continuous map if and only if for every open set U in Y, f-'(U) is open in X. Definition. Let f : X → Y be a function between topological spaces X and Y, and let x e X. Then f is continuous at the point x if and only if for every open set V containing f(x), there is an open set U containing x such that f(U) c V. Thus a function f : X → Y is continuous if and only if it is continuous at each point. AU B, where A, B are closed in X. Let ƒ : Theorem 7.10 (Pasting lemma). Let X = A → Y and g : B → Y be continuous functions that agree on An B. Then the function h: AUB → Y defined by h = f on A and h = g on B is continuous.
Transcribed Image Text:Definition. Let X and Y be topological spaces. A function or map f : X → Y is a continuous function or continuous map if and only if for every open set U in Y, f-'(U) is open in X. Definition. Let f : X → Y be a function between topological spaces X and Y, and let x e X. Then f is continuous at the point x if and only if for every open set V containing f(x), there is an open set U containing x such that f(U) c V. Thus a function f : X → Y is continuous if and only if it is continuous at each point. AU B, where A, B are closed in X. Let ƒ : Theorem 7.10 (Pasting lemma). Let X = A → Y and g : B → Y be continuous functions that agree on An B. Then the function h: AUB → Y defined by h = f on A and h = g on B is continuous.
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