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?