Show if $f_1|_{\mathbb Q} = f_2|_{\mathbb Q}$ then $f_1 = f_2$

Question

Let $f_1,f_2: \mathbb R \to \mathbb R$ be two continuous functions. I want to show that if $f_1|_{\mathbb Q} = f_2|_{\mathbb Q}$ then $f_1 = f_2$.

How could I show this?

Answer 1

$|f_1(x) - f_2(x)| = |f_1(x) - f_1(q) + f_2(q) - f_2(x)| $ where $q$ is rational sufficiently close to $x$. Apply triangle inequality.