Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Prove that if $S$ is a dense (i.e., closure of $S$ is $\mathbb{R}$) subset of $\mathbb{R}$ such that $f(s) = 0$ for every $s$ in $S$, then $f(x) = 0$ for every $x$ in $\mathbb{R}$.
I think the following may be helpful:
Let $f$ be a function with domain $E$ and fix $P\in E$. The function $f$ is continuous at $P$ if and only if for every sequence $\{a_{j}\}\subseteq E$ satisfying $\lim_{j \to \infty} a_{j} = P$ it holds that $\lim_{j \to \infty} f(a_{j}) = f(P)$