Is the Graph of a Continuous Function a Closed Set?

[Ted123]

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]

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]

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]

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?