Is the Graph of a Continuous Function a Closed Set?

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Homework Help Overview

The discussion revolves around the properties of the graph of a continuous function, specifically whether it constitutes a closed set in the context of Euclidean space. The subject area involves concepts from topology and real analysis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore definitions of closed sets and consider geometric interpretations of continuous functions. There are inquiries about how to demonstrate that the graph is closed in \(\mathbb{R}^2\). One participant suggests examining the function defined by the difference between the function value and the y-coordinate.

Discussion Status

The discussion is active with participants questioning the definitions and exploring different approaches to the problem. There is no explicit consensus yet, but various lines of reasoning are being examined.

Contextual Notes

Participants are discussing the implications of continuity and the characteristics of closed sets, with some focusing on the geometric representation of the graph in the plane.

Ted123
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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?
 
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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
 
lanedance said:
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
 
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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