How do I show a function on 2-adic units is continuous?

Solution 1:

As for the map

$$f: x \mapsto \frac{3x+1}{2^{v_2(3x+1)}} = (3x+1)\cdot |3x+1|_2,$$

it is the composition of $x\mapsto 3x+1$ and $y\mapsto y |y|_2$, so we want to enquire where these are continuous, the only interesting part being actually the absolute value map $| \cdot|_2$ itself. Viewed as map $(\Bbb{Q}_2, |\cdot|_2)\rightarrow (\Bbb{Q}_2, |\cdot|_2)$, the absolute value is not continuous at $0$ (because $|2^n|_2 =2^{-n}$ does not converge $2$-adically for $n\to \infty$), but outside of $0$, it is actually locally constant and hence continuous. So the composite map $g: (\Bbb{Q}_2, |\cdot|_2)\rightarrow (\Bbb{Q}_2, |\cdot|_2)$ is continuous everywhere except at $x=-\frac{1}{3}$. (Note, however, that this point $-1/3$ w.r.t. the $2$-adic metric does lie in every neighbourhood of $\mathbb{N}$, even in every neighbourhood of the odd natural numbers, as mentioned here recently.)

With a similar argument, the function $\tilde f$ in your answer -- which, if I understand it correctly, is nothing else than $x\mapsto |x|_2^{-1}\cdot f(x\cdot |x|_2)$ -- is continuous as function $(\Bbb{Q}_2, |\cdot|_2)\rightarrow (\Bbb{Q}_2, |\cdot|_2)$, except at the points $-\frac{2^k}{3}, k \in \mathbb{Z}$.

As for the function $g$, which I would rewrite as $x\mapsto (3x+|x|_2^{-1})\cdot |3x+|x|_2^{-1}|_2$, it looks as if it is continuous except at $0$ and all $-\frac{2^k}{3}, k \in \mathbb{Z}$.