prove that for a continious function if f(q) = 0 for all q where q is rational then all f(x) = 0 when x is real

Question

it is given that f is Continious. and $\forall q\in\mathbb{Q}$
$f(q)=0$

how do I prove that:
for all $x\in\mathbb{R}$ : $f(x) = 0$

Answer 1

For every real number $\alpha$ there is a sequence $\alpha_1 , \alpha_2, ...$ of rational numbers such that $\lim_{n\to\infty}\alpha_n = \alpha $. Let $\alpha$ be irrational, since we know it's true for rationals. Then, since $f$ is continuous,

$f(\alpha) = f(\lim_{n\to\infty} \alpha_n) = \lim_{n\to\infty} f(\alpha_n) = 0$