Defining a ring homomorphism from $\mathbb{C}\left [ x_1, \dots, x_n \right ]/\left ( f_1, \dots, f_r \right )$ I am working on a problem in Artin's Algebra related to the algebraic geometry talked in Chapter 11. The problem number is 9.2., F.Y.I.
Here goes the problem:

Let $f_1, \dots, f_r$ be complex polynomials in the variables $x_1, \dots, x_n$, let $V$ be the variety of their common zeros, and let $I$ be the ideal of the polynomial ring $R = \mathbb{C}\left [ x_1, \dots, x_n \right ]$ they generate. Define a ring homomorphism from the quotient ring $\bar{R} = R/I$ to the ring $\mathcal{R}$ of continuous, complex-valued functions on $V$.

I attempted to use the correspondence theorem w.r.t. the variety of a set of polynomials, i.e. the maximal ideals bijectively correspond to the point in $V$ and we may somehow define the continuous functions there. However I cannot come up with any idea further. Also, the term 'continuous' here seems redundant since I expect the homomorphism will carry polynomials to polynomials. 
I appreciate your participation and will be thankful to anything from hints to full solution. 

It's easier than it looks.  To each element $h + I$ in $R/I$, associate the function $\phi: V \rightarrow \mathbb{C}$ by setting
$$\phi(t_1, ... , t_n) = h(t_1, ... , t_n)$$
for every $(t_1, ... ,t_n)$.  Check that this doesn't depend on the specific coset representative you use.  
