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-   -   Closed set (metric spaces) (http://www.physicsforums.com/showthread.php?t=548254)

 Ted123 Nov7-11 02:39 PM

Closed set (metric spaces)

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a continuous function (standard metric).

Show that its graph $\{ (x,f(x)) : x \in \mathbb{R} \}$ is a closed subset of $\mathbb{R}^2$ (Euclidean metric).

How to show this is closed?

 lanedance Nov7-11 03:01 PM

Re: Closed set (metric spaces)

what are your definitions of closed?

thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2

 Ted123 Nov7-11 03:14 PM

Re: Closed set (metric spaces)

Quote:
 Quote by lanedance (Post 3603566) what are your definitions of closed? thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2
Well a set $A$ is closed if $\partial A \subset A$, i.e. $\partial A \cap A^c = \emptyset$

 Ted123 Nov7-11 03:17 PM

Re: Closed set (metric spaces)

How could I show it is closed by considering the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y)=f(x)- y$?

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