# Closed set (metric spaces)

1. ### Ted123

446
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?

2. ### lanedance

3,307
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

3. ### Ted123

446
Well a set $A$ is closed if $\partial A \subset A$, i.e. $\partial A \cap A^c = \emptyset$

4. ### Ted123

446
How could I show it is closed by considering the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y)=f(x)- y$?

Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?