Suppose [itex]f:\mathbb{R}\to \mathbb{R}[/itex] is a continuous function (standard metric). Show that its graph [itex]\{ (x,f(x)) : x \in \mathbb{R} \}[/itex] is a closed subset of [itex]\mathbb{R}^2[/itex] (Euclidean metric). How to show this is closed?
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 [itex]A[/itex] is closed if [itex]\partial A \subset A[/itex], i.e. [itex]\partial A \cap A^c = \emptyset[/itex]
How could I show it is closed by considering the function [itex]f : \mathbb{R}^2 \to \mathbb{R}[/itex] defined by [itex]f(x,y)=f(x)- y[/itex]?