Let $S=C[0,1]$ be the set of real-valued continuous functions defined on the closed interval $[0,1]$, where we define $f+g$ and $fg$, as usual, by $(f+g)(x)=f(x)+g(x)$ and $(fg)(x)=f(x)g(x)$. Let $0$ and $1$ be the constant functions $0$ and $1$, respectively. Show that $S$ has no idempotents (except $0$ and $1$).
I know that an idempotent is an element $e$ such that $e^2=e$. I am not sure where to start with this question. Any help is much appreciated!
Hint: If $f^2 = f$, then for all $x \in [0,1]$ $$f(x)\big(f(x) - 1\big) = 0$$ Now use the facts that $f(x) \in \mathbb R$ and $\mathbb R$ is an integral domain, and that $f$ is continuous.