A Hilbert curve is a continuous space-filling curve that maps $\mathbb{R}^n$ to $\mathbb{R}$. 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 $\mathbb{CN}^n \mapsto \mathbb{CN}$ where $\mathbb{CN}$ denotes the set of computable numbers.

At first glance, this seems like a true statement because we have an algorithm that takes any $t \in \mathbb{CN}$, iterates over its binary digits, and sequentially produces the binary digits of all numbers in its corresponding (unique) tuple $T \in \mathbb{CN}^n$. 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.

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