Let $f$ the function defined on $\mathbb Q$ by : $ f(n/m) = n $.
I would like to know whether it is true that : $ \forall q\in \mathbb Q - \{ 0 \} \quad \forall R>0 \quad \exists \delta >0\quad \forall x\in \mathbb Q \quad :\quad 0<|x-q|<\delta \Rightarrow |f(x)|>R $
(That is : $ \lim_{x \to q, x \in \mathbb Q} f(x) = \infty$, for all rational $q ≠ 0$).
Thanks in advance !
You need to clarify that $x \neq q$ when you are quantifying over all $x$, and that fractions are always given in lowest terms. If you do that, then it is true.
Lemma: If $x = m/n \neq q$, where $q = r/s$, then $|x - q| \geq 1/ns$.
Corollary: If $\delta < 1/ns$ for some positive integer $n$, and $|x - q| < \delta$ for $x$ rational, then $x = m'/n'$ in lowest terms where $|n'| > n$.
Lemma: If $|x - q| < \delta$ where $q \neq 0$ and $x = m'/n'$, then $|m'| \geq (|q| - \delta)|n'|$.
Corollary: $|f(x)| \to \infty$ as $|x - q| \to 0$ for $x \neq q$.