Closed set (metric spaces)

  1. 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?
  2. jcsd
  3. lanedance

    lanedance 3,307
    Homework Helper

    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
  4. 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]
  5. 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]?
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