Help prove $f(x)= 0$ for all $x$ when the set $S$ in $\mathbb{R}$ is dense

Question

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ has the property that $f$ is continuous and $f(x) = 0$ for all $x\in S$, where $S\subseteq\mathbb{R}$ is dense. Prove that $f(x) = 0$ for all $x\in\mathbb{R}$.

Answer 1

Use the fact that $\{x\in\mathbb{R}| f(x) = 0\}$ is closed.

Answer 2

Hint: For any real $x$ you can choose a sequence $s_n \to x$ where $s_n \in S,$ since $S$ is assumed dense.