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Exercise 2. Let f: X→Y be a map between the topological spaces X and Y. a) Let V V(x) be a neighborhood of x E X. If the restriction fVz: V₂ →Y is continuous at x, show that f: XY is continuous at x.

Exercise 2. Let f: X→Y be a map between the topological spaces X and Y. a) Let V V(x) be a neighborhood of x E X. If the restriction fVz: V₂ →Y is continuous at x, show that f: XY is continuous at x.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 2
that J is automatically continuous? true prove it, II false give an examplej Exercise 2. Let f: X→Y be a map between the topological spaces X and Y. a) Let V V(x) be a neighborhood of x E X. If the restriction f Va Va →Y is continuous at x, show that f: XY is continuous at x. b) Accu Fuengigo 2
Transcribed Image Text:that J is automatically continuous? true prove it, II false give an examplej Exercise 2. Let f: X→Y be a map between the topological spaces X and Y. a) Let V V(x) be a neighborhood of x E X. If the restriction f Va Va →Y is continuous at x, show that f: XY is continuous at x. b) Accu Fuengigo 2
Expert Solution
Step 1

assume that for each x∈X, there is a neighborhood Vx of x

s.tsuch that f|Vx is continuous at x

Claim:f:X-Y  is conitnuous at x

Let x be any point in X. Let V be any open set in Y which contains f(x).

By assumption, there is a neighborhood W of x such that f|W is continuous

Since (f|W)-1(V) is an open set in W, there is an open set Vx in X such that

(f|W)-1(V)=Vx∩U

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