A Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if a Hilbert curve becomes a continous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

Note: The specific curve in question, along with the algorithms I mention above, is described here.

Thanks in advance for any comments, ideas and pointers to the relevant literature.