Suppose that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $g(x)=0$ if $x$ is rational. Prove that $g(x)=0$ for all $x$ in $\mathbb{R}$.
Proof:
Let $\{x_n\}$ be a sequence of $\mathbb{Q}$, then the limit of $\{x_n\}$ is in $\mathbb{Q}$ and $\lim\limits_{n\rightarrow\infty}f(x_n)=f(x)=0$ where $x\in\mathbb{Q}$. And by that fact that $\mathbb{Q}$ is dense of $\mathbb{R}$, so the limit of $\{x_n\}$ is in $\mathbb{R}$. Thus, since the function $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous, we can conclude that $g(x)=0,\forall x\in\mathbb{R}$.
It seems I just put everything into my proof to convince me that is right. Can anyone check my solution? Thanks