Let $R$ denote the ring of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, where addition and multiplication are given, respectively, by $(f + g)(x) = f(x) + g(x)$ and $(fg)(x) = f(x)g(x)$ for every $f, g \in R$ and $x \in \mathbb{R}$. A zero-divisor in $R$ is a non-zero $f \in R$ such that $fg = 0$ for some non-zero $g \in R$. Pick the true statement(s) from below: (A) $R$ has zero-divisors. (B) If $f$ is a zero-divisor, then $f^{2} = 0$. (C) If $f$ is a non-constant function and $f^{-1}(0)$ contains a non-empty open set, then $f$ is a zero-divisor. (D) $R$ is an integral domain.
Let $R$ denote the ring of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, where addition and multiplication are given, respectively, by $(f + g)(x) = f(x) + g(x)$ and $(fg)(x) = f(x)g(x)$ for every $f, g \in R$ and $x \in \mathbb{R}$. A zero-divisor in $R$ is a non-zero $f \in R$ such that $fg = 0$ for some non-zero $g \in R$. Pick the true statement(s) from below:\\
(A) $R$ has zero-divisors.\\
(B) If $f$ is a zero-divisor, then $f^{2} = 0$.\\
(C) If $f$ is a non-constant function and $f^{-1}(0)$ contains a non-empty open set, then $f$ is a zero-divisor.\\
(D) $R$ is an integral domain.