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Exercise 2 (-8). Assume (X, dx) and (Y, dy) are metric spaces. Show that a map f: X → Y is continuous at x EX if and only if for each > 0, we can find a 8 >0 such that dx(x',x) < 8, for x' E X, implies dy(f(x'), f(x)) < €.

Exercise 2 (-8). Assume (X, dx) and (Y, dy) are metric spaces. Show that a map f: X → Y is continuous at x EX if and only if for each > 0, we can find a 8 >0 such that dx(x',x) < 8, for x' E X, implies dy(f(x'), f(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 (-8). Assume (X, dx) and (Y, dy) are metric spaces. Show that a map f : X → Y is continuous at x = X if and only if for each e > 0, we can find a 8 >0 such that dx(x',x) < 8, for x' E X, implies dy(f(x'), f(x)) < €. Exercise 3. Prove the following theorem: Theorem Let f Y a man between the tonological spaces X and Y The following statements
Transcribed Image Text:Exercise 2 (-8). Assume (X, dx) and (Y, dy) are metric spaces. Show that a map f : X → Y is continuous at x = X if and only if for each e > 0, we can find a 8 >0 such that dx(x',x) < 8, for x' E X, implies dy(f(x'), f(x)) < €. Exercise 3. Prove the following theorem: Theorem Let f Y a man between the tonological spaces X and Y The following statements
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