continuity – Is the Collatz function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ continuous on $Bbb Z_2^times$?

Is the Collatz function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ continuous on $Bbb Z_2^times$?

Let $Bbb Z_2^times$ be the 2-adic units.

Then e.g. $3mapstodfrac{10}2=5$

Attempt

Here’s what I know: The function above can be thought of as a map $xmapsto3x+2^{nu_2(x)}$ on all of $Bbb Z_2toBbb Z_2$ which descends to the quotient map $xmapsto{2^ixinBbb Z_2:iinBbb Z}$.

Since $xmapsto3x+2^{nu_2(x)}$ is continuous and the function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ descends to the quotient, it must too be continuous on $Bbb Z_2/{sim}$

However, I think $Bbb Z_2/{sim}$ isn’t homeomorphic to $Bbb Z_2^times$. If it were, I could answer in the positive. There’s an obvious bijection between them but I think $Bbb Z_2/{sim}$ has the trivial topology because every element is arbitrarily close to $0$. So the trivial continuity in $Bbb Z_2/{sim}$ says nothing of continuity in $Bbb Z_2^times$.