Intermediate value theorem/Not for Q/Exercise/Solution

The function

$f\colon \mathbb {Q} \longrightarrow \mathbb {Q} ,x\longmapsto x^{2}-2,$

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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