# real analysis – \$∀x∈mathbb Q\$, \$f(x)≧0\$ implies \$∀x∈mathbb R\$, \$f(x)≧0\$

Let $$f(x)∈mathbb R(x)$$ be polynomial over real number.

I think the following claim holds.

Claim

$$∀x∈mathbb Q$$, $$f(x)≧0$$ implies $$∀x∈mathbb R$$, $$f(x)≧0$$

I think this holds because of density of $$mathbb Q$$ in $$mathbb R$$.

My question

1. Does this claim correct?
2. Can we extend this statement to some general results?