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+----- Thread: An interesting family of functions (/showthread.php?tid=2319)
An interesting family of functions - mklabgr - 10-01-2026
The post starts from base-$b$ expansions for an integer $b > 1$. Terminating expansions have a second representation, which comes from the identity $1 = \sum_{k=1}^{\infty} (b-1),b^{-k}$, so $1 = 0.999\ldots$ in decimal. To remove this ambiguity, the author fixes a canonical expansion $x = x_n x_{n-1}\ldots x_0.x_{-1}x_{-2}\ldots$ with $x = \sum_{i \le n} x_i b^i$ and $x_i \in {0, \ldots, b-1}$. Two conditions apply: if $x_n = 0$ then $n = 0$, and for every $i$ there is a $k > i$ with $x_{-k} \neq b-1$.
A number $x$ terminates exactly when $x = 0$ or $x = p/q$ in lowest terms, where every prime factor of $q$ divides $b$. The base conversion functions $B_{b,c}$ for $c \neq 0$ then read the base-$b$ digits as if they were base-$c$ digits: $B_{b,c}(x) = \sum_{i \le n} x_i c^i$ whenever the sum converges. Clearly $B_{b,c}(0) = 0$ and $B_{b,c}(1) = 1$. For an integer $N$ and $0 \le y < 1$, $B_{b,c}(N + y) = B_{b,c}(N) + B_{b,c}(y)$, so only integers and $[0,1]$ matter.
For the domain, convergence requires the terms $x_i c^i \to 0$ as $i \to -\infty$. If $|c| \le 1$, this forces the digits to be eventually zero, so the domain is exactly the numbers with terminating expansions. If $|c| > 1$, the series converges for all $x \in \mathbb{R}$, since the digits are bounded by $b-1$, and the size of $B_{b,c}(x)$ is controlled using $n \le \log_b x = \ln x / \ln b$.
For continuity, $B_{b,b} = \mathrm{id}$ is trivially continuous. At $0$, $B_{b,c}$ is continuous if and only if $|c| > 1$: for $|c| \le 1$ the sequence $b^{-n} \to 0$ gives $B_{b,c}(b^{-n}) = c^{-n} \not\to 0$. The main result is that for $|c| \le 1$, $B_{b,c}$ is discontinuous at every point of its domain. For $|c| > 1$, it is continuous at every non-terminating $x$ and discontinuous at every terminating one. The proof for non-terminating $x$ uses the fact that nearby $y$ share the first $N$ digits with $x$, with digit differences satisfying $-(b-1) \le x_i - y_i \le b-1$.
For directional continuity, if $|c| \le 1$ the function is both left and right discontinuous on its domain. If $|c| > 1$, it is left-continuous everywhere but right-continuous only at non-terminating points. At a terminating $x = x_n \ldots x_0.x_{-1}\ldots x_{-m}000\ldots$ with $x_{-m} \neq 0$, the left limit satisfies $\lim_{y \to x^-} B_{b,c}(y) = B_{b,c}(x) - c^{-m}$, a jump discontinuity of size $c^{-m}$. The post closes with $B_{2,10}\colon [0,1] \to [0,1]$, which writes $x$ in binary and reads it as a decimal. It has a jump at every dyadic rational $p/2^n$, a dense set, yet that set is countable, so the function is Riemann-integrable. Its image is Cantor-set-like, which made the author's Python plot uninteresting, and readers are invited to produce better plots.
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