(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [itex]f:\mathbb{R}\to\mathbb{R}[/itex] and [itex]g:\mathbb{R}\to\mathbb{R}[/itex] are continuous functions show that:

(a) the graph of [itex]f[/itex], [itex]\{(x,f(x)) : x\in\mathbb{R} \}[/itex] is a closed subset of [itex]\mathbb{R}^2[/itex].

(b) [itex]\{ (x,f(x),g(x)) : x\in \mathbb{R} \}[/itex] is a closed subset of [itex]\mathbb{R}^3[/itex].

3. The attempt at a solution

I've done (a): the graph can be written as [itex]\{ (x,y) \in \mathbb{R}^2: y-f(x) = 0 \}[/itex] so we can use preimages:

Considering the function [itex]F : \mathbb{R}^2 \to \mathbb{R}[/itex] defined [itex]F(x,y) = y-f(x)[/itex]; [itex]F[/itex] is continuous and the graph of [itex]f[/itex] is the preimage [itex]F^*(0)[/itex] and since [itex]\{0\}[/itex] is closed so is the graph.

(b) must be similar but I can't see how to write the set in a form where I can use preimages immediately.

The set can be written as:

[itex]\{ (x,y,z)\in\mathbb{R}^3 : y = f(x) , z = g(x) \}[/itex]

i.e. [itex]\{ (x,y,z)\in\mathbb{R}^3 : y - f(x) = g(x) - z = 0 \}[/itex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Closed set (metric spaces)

**Physics Forums | Science Articles, Homework Help, Discussion**