Proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous

Question

Could you tell me some webpages or books where I can find the proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous. I know how it can proof if $P:\mathbb{R}\to\mathbb{R}$, but I don't know how it works if $P:\mathbb{R}^n\to\mathbb{R}$.

Answer 1

If you want to prove that every polynomial map $\mathbb R^n\to\mathbb R$ is continuous, it is not a good idea to consider a general polynomial and give an $\varepsilon$-$\delta$-proof. Instead, you should start with the building blocks, i.e. the projections $x\mapsto x_i$. Use an $\varepsilon$-$\delta$-proof to show that all the projections are continiuous. Now tackle products and sums: Given continuous maps $f,g\colon \mathbb R^n\to\mathbb R$ and $\lambda\in\mathbb R$, show that $f+g$, $f\cdot g$ and $\lambda f$ given by pointwise addition and pointwise multiplication are continuous. Again, this is done by $\varepsilon$-$\delta$-arguments. Now, by induction, any finite sums or products of continuous functions are continuous. Hence, the monomial maps are continuous as finite products of projections and polynomial maps are continuous as finite sums of continuous functions.