What are the necessary conditions for a closed subset in metric spaces?

[Ted123]

Homework Statement

If $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ are continuous functions show that:

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

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

The Attempt at a Solution

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

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

$\{ (x,y,z)\in\mathbb{R}^3 : y = f(x) , z = g(x) \}$

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

 

[canis89]

Hint: What must be the values of $a,b\in\mathbb{R}$ so that $$a^2+b^2=0?$$