Intermediate value theorem/Not for Q/Exercise/Solution

The function

$f:\mathbb{Q}⟶\mathbb{Q},x⟼x^2-2,$

is continuous and we have $f(0)=-2<0$ and $f(2)=2>0$. If the Intermediate value theorem would also hold for the rational numbers, then there would be inside the rational interval $[0,2]$ a zero for this function, so there would be a $q\in \mathbb{Q}$ with $q^2=2$. But this is not possible, since the square root of $2$ is irrational.

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