Question

Let f : X → Y be a function and Y be compact Hausdorff. Then f is continuous if and only if the graph of f , G f = {(x, f (x)) : x ∈ X} is closed in X × Y .

Name: let f x y be a function and y be compact hausdorff then f is continuous if and only if the graph of f g f x f x x x is closed in x y 35731
Uploaded: 2023-03-08T18:32:07-08:00
Duration: 1 min 33 s
Channel: Sri K
Description: let f x y be a function and y be compact hausdorff then f is continuous if and only if the graph of f g f x f x x x is closed in x y 35731

          Let f : X → Y be a function and Y be compact Hausdorff.
Then f is continuous if and only if the graph of f , G f = {(x, f
(x)) : x ∈ X}
is closed in X × Y .

Added by Nathan C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/08/2023

Step 1

Assume f is continuous. Let's show that Gf is closed in X × Y. To do this, we need to show that the complement of Gf in X × Y is open. Let (x, y) be a point in (X × Y) - Gf, which means that y ≠ f(x). Since Y is Hausdorff, there exist disjoint open neighborhoods Show more…

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Let f : X → Y be a function and Y be compact Hausdorff. Then f is continuous if and only if the graph of f , G f = {(x, f (x)) : x ∈ X} is closed in X × Y .