I was going through chapter 10 in Artin there I found following proposition -
Here I am unable to figure out how the homomorphism function will look like in from Real no.'s to Ring of real valued continuous functions coz as mentioned in first line of proof substitution principle is used to show the map from real polynomial R[x] to real valued polynomial function. I was thinking it to be as -
$$\phi(1) = identitymap$$ $$\phi(0) = zeromap$$ $$\phi(k) = \phi(1+1+...ktimes) = k*identity map$$
Is it correct? Also isn't it an isomorphic map from R[x1,..xn] to R??
Artin is claiming that if you map every formal polynomial $p(x_1,\ldots ,x_n)$ in $n$ variables with real coefficients to the function
$ f_p:\mathbb{R}^n\rightarrow\mathbb{R}, (a_1,\ldots ,a_n)\mapsto p(a_1,\ldots ,a_n) $
then this maps $\mathbb{R}[x_1,\ldots ,x_n]$ injectively into the ring $R$ of real valued continous functions $\mathbb{R}^n\rightarrow\mathbb{R}$. Of course this map is not surjective, because there exist non-polynomial continous functions.