I read about Conway's base 13 function and felt encouraged to procrastinate on my homework and play around with functions that involve binary expansions. This one function $K$ I came across caught my attention.

Let $t \in (0,1)$. Taking the binary expansion, we can write $t= \sum_{n=1}^\infty \frac{a_n}{2^n}$, where $a_n \in \{0,1\}$ and $(a_n)$ cannot have an endless tail of 1's (i.e. for any $N \in \mathbb{N}$ with $a_N=1$, there exists some $n \geq N$ with $a_n =0$). This makes the binary expansion unique for any $x \in (0,1)$.

We can create 2 numbers from this: $x= \sum_{n=1}^\infty \frac{a_{2n-1}}{2^n}$ and $y= \sum_{n=1}^\infty \frac{a_{2n}}{2^n}$. We say $K(t)=(x,y)$.

In other words, we convert $t$ to binary format (e.g. $0.3141... \rightarrow 0.0101...$), and put all the odd-indexed digits in the binary expansion of $x$, and even-indexed digits in the binary expansion of $y$. Then you convert $x,y$ back to decimal format. Plotting the path, we have:

K function

This looks like a space-filling curve! Neat. It oddly looks similar to the Hilbert Curve. My question is if this function is continuous, since it zig-zags so much about the rational numbers.

Just for funzies, when I add $x$ and $y$, I can create a new function $k(x):=x+y$. Doing so, I get this neat looking graph:

k function

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    $\begingroup$ I'm happy to share my code if anyone wants it. $\endgroup$ – Spencer Kraisler Nov 20 '19 at 7:48

This function is not continuous: its value at $t=\frac12$ is $(\frac12,0)$, but its value at numbers just less than $\frac12$ is very close to $(\frac12,1)$. Analogous discontinuities occur at every value $t=\frac a{2^k}$.

For the record, this function is "space-filling", in that its range is $[0,1]\times[0,1] \setminus \{(1,1)\}$. (Just pick the desired target coordinates and interleave their bits to find the input $t$ that gets mapped to the desired point. The fact that $(1,1)$ is omitted is because we need to use $1=0.111\dots$ to achieve that coordinate, and we can arrange for this to happen for either coordinate but not for both separately.)

  • $\begingroup$ So it’s discontinuous on the dyadics. I wonder if it’s continuous everywhere else. $\endgroup$ – Spencer Kraisler Nov 20 '19 at 8:13
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    $\begingroup$ It is discontinuous on every terminating decimals of base 2, and continuous elsewhere, since every infinite decimals of base 2 can only be approximated by decimals with identical first $n$th decimal place. But terminating fractions like $0.1$ can be approximated by $0.011111...$. $\endgroup$ – Local Kleinian Manifold Nov 20 '19 at 8:44
  • $\begingroup$ That makes sense. Great deduction. How did you figure thag out? Any other cool info you have about this? $\endgroup$ – Spencer Kraisler Nov 20 '19 at 9:00

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