# Thread: map

1. ## map

Suppose $f(x) = \begin{cases} 0 \ \ \ \text{if} \ \ x \notin \mathbb{Q} \\ 1/q \ \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = p/q \ \text{in lowest terms} \end{cases}$

1. Fix $t \in \mathbb{R}$ and $n \in \mathbb{N}$. Show that $f$ maps only finitely many elements of $(t-1/2, t+1/2)$ to $1/n$.

So the length of this interval is $1$. I think its $n-1$. But that is intuition. How do you make it rigorous?

2. Prove that $f$ is continuous at every irrational number. So $\forall \epsilon >0, \exists \delta >0$ such that $|x-a| < \delta \implies |f(x)| < \epsilon$. Do we want to elements that dont map to $1/n$?

2. Basically we are trying to show that $f$ maps a finite number of rational numbers in lowest terms in the interval $I = (t- \frac{1}{2}, t+\frac{1}{2})$ to $\frac{1}{n}$. Let $x = p/n \in I$ and in lowest terms. Then $p$ and $n$ are coprime. Can we somehow use this to deduce that the maximum number is $n-1$? number of fractions $p/n$ in interval is $n$. number in lowest terms is $\leq n-1$.