Proving a function is continuous by writing it as composition of continuous functions

Question

Given the following functions

$$ f:R^3 \rightarrow R^2, \hspace{20pt} f(x_1,x_2,x_3) = ((x_1)^2+x_1x_3, \ (x_2)^2+x_1x_3)$$

$$g:R^9 \rightarrow R, \hspace{20pt} g(A)=|x_{11} + x_{22} + x_{33}| \ \text{where A is real}\ 3\times3 \ \text{matrix}$$

I need to prove that they are continuous w.r.t the corresponding euclidean metric by writing them as composition of continuous functions. Problem is I find it very confusing when it's a map between different dimensions so not sure how those functions need to look.

Answer 1

You need:

1. Projections $p_i:x\mapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
2. If you have 2 continuous functions $f,g:R\to R$ then $(f,g):R^2\to R^2; (x,y)\mapsto (f(x),g(y))$ is continuous.
3. Also $+:R^2 \to R; (x,y)\mapsto x+y$ and $M:R^2 \to R; (x,y)\mapsto x\cdot y$ are continuous.

So you can define (spoiler):

$f(x_1,x_2,x_3)= (+(M\circ(p_1,p_1),M\circ(p_1,p_3)), +(M\circ(p_2,p_2),M\circ(p_1,p_3)) ) (x_1,x_2,x_3)$

For g you also need the absolute value function $|\cdot|$ to be continuous.