Properties of dense sets

Question

Let $A$ be a countable set dense in $[0,1]$ and we have $f(n)=0$ for all $n$ in $A$.

Does this imply that $f(x)=0$ almost everywhere in $[0,1]$?

Answer 1

If $f$ is also continuous, then $f$ has to vanish everywhere, since, for every $x\in [0,1]$, there exists a $x_n\to x$, with $\{x_n\}\subset A$, and $$0=f(x_n)\to f(x).$$