Hello!!!

So much homework :-\

I need to show that every maximal ideal in the ring R, which is made of all continuous functions [0,1]-->R, is of the form M(c)={f in R| f(c)=0).

I think I have a kinda of a topological proof formed, and of course it can be specifically used here (the proof is for any hausdorff-compact space), but I'm seeking a more algebraic proof, which I'm sure exists since most of us haven't studied topology yet.

I was to show that if M is a maximal ideal in R, then there must exist a point c in [0,1] which satisfies f(c)=0 for all f in M. Therefore, I start by contradically assuming that for all x in [0,1] there exists a function f_x in M such that f_x(x)=0. I cannot reach a contradiction though...

Thank you in advance...

Tomer.