Continuity of polynomials of two variables

Question

I'm working on a proof of the following statement (from T.Tao's Analysis 2 book):

"Let $n,m\geq 0$ be integers and suppose that for every $0\leq i\leq n$ we have a real number $c_{ij}$. Form the function $P\colon\mathbb{R}^2\to\mathbb{R}, P(x,y):=\sum_{i=0}^n\sum_{j=0}^m c_{ij}x^i y^j$.

(1) Show that $P$ is continuous. (DONE)

(2) Conclude that if $f\colon X\to\mathbb{R}$ and $g\colon X\to\mathbb{R}$ are continuous functions then the functions $P(f,g)\colon X\to\mathbb{R}$ defined by $P(f,g)(x):= P(f(x),g(x))$ is also continuous."

I've managed to prove part (1) and tried to prove part (2) using the $\varepsilon -\delta$ definition of continuity (which is the definition used by the text) with no success. So, I'd appreciate any hint about how to prove this second part of the statement.

Best regards,

lorenzo.

Answer 1

Claim (2) follows from first principles about continuity.

If the "component functions" $f:\>X\to{\mathbb R}$ and $g:\>X\to{\mathbb R}$ are continuous then the map $$\psi:\quad X\to{\mathbb R}^2,\qquad x\mapsto\bigl(f(x),g(x)\bigr)$$ is continuous. Now the function $$P(f,g): \quad X\to{\mathbb R},\qquad x\mapsto P\bigl(f(x),g(x)\bigr)$$ is nothing else but $P\circ\psi$, hence is continuous by claim (1) and the principle that the composition of continuous maps is again continuous.

Answer 2

Let $x_0 \in X $ and $x_n \rightarrow x_0$ in $X$ then from continuity of $f,g$ we have that $$g(x_n) \rightarrow g(x_0)$$ $$f(x_n) \rightarrow f(x_0)$$

Also $v_n=(f(x_n),g(x_n)) \in \mathbb{R}^2$.

Prove that $v_n=(f(x_n),g(x_n)) \rightarrow v_0=(f(x_0),g(x_0))$ in $\mathbb{R}^2$

Then use the continuity of $P$ in $\mathbb{R}^2$ which you proved in the first part of the exercise.

If you want hints of an epsilon-delta proof then let me know.