# Is this very weird function continuous?

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:

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:

• I'm happy to share my code if anyone wants it. – 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.)
• 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 $0.011111...$. – Local Kleinian Manifold Nov 20 '19 at 8:44