# 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 $$Bbb Z_2^times$$.