Exercise 3Part 4 ### Mathematical Problem Statement**Problem 4:**If \( X \) is a non-empty Baire space, which is Hausdorff and without isolated points, and if \((Y, d)\) is a metric space with a mapping \( f: X \to Y \), show that:\[\text{Cont}(f) = \{ x \in X \mid f \text{ is continuous at } x \}\]The set of points in \( X \) at which \( f \) is continuous cannot be dense and countable.

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Answered: 4) If X is a non-empty Baire space,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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4) If X is a non-empty Baire space, which is Hausdorff and without isolated points. If (Y, d) is a metric space and f: X→Y is a map, show that Cont (f) = {x X| f is continuous at X}, the set of points in X at which is continuous, cannot be dense and countable.