Analyzing an explicit function whose graph is "extremely scattered" across $\mathbb{R}\times\mathbb{R}$?

Question

Question: Are there any special properties of the following function?

Consider the function $f:\mathbb{R}\to\mathbb{R}$, where for a base-3 expansion of $x\in\mathbb{R}$, we take the "pseudo-random" iterations of function $\mathscr{F}:\{0,1,2\}\to\{0,1,2\}$ where for the first iteration, replace base-3 digit zero with one; one with two, and two with zero.

In mathematical terms, if $\mathscr{F}:=\left\{(0,1),(1,2),(2,0)\right\}$, such that equation: $$\mathscr{F}^{k}=\underset{k\text{ times}}{\underbrace{\mathscr{F}(\mathscr{F}(\cdots\mathscr{F}(x)))}}$$ is the $k$-th iteration of $\mathscr{F}$, we define a function with "pseudo-random" outputs in $\mathbb{N}$. If we define $g_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ where:

$$g_{\varepsilon}(x)=\begin{cases} \quad\!\!(2\varepsilon)^{x/(2\varepsilon)} & \quad\!\! x<-\varepsilon\\ -1/x & \!-2\varepsilon\le x <0\\ \quad \! 0 & \!\!\quad x=0 \\ \quad\!{1}/{x} & \quad\!\! 0<x<\varepsilon \\ \quad\!\!\,\varepsilon^{-x/\varepsilon} & \quad\!\! x\ge \varepsilon\\ \end{cases}$$

such that $\left[\cdot\right]$ rounds to the nearest integer, then function $z_{\varepsilon}:\mathbb{R}^2\to\mathbb{R}$ or

$$z_{\varepsilon}(x,k)=\big[g_{\varepsilon}(x) k\sin(g_{\varepsilon}(x)k)\big]$$

contains "pseudo-random" outputs of $\mathbb{N}$, where furthermore:

$$\small{f_{\varepsilon}(x)=\left\{\sum\limits_{k=-\infty}^{\infty}{\text{sign}\left(\mathscr{F}^{z_{\varepsilon}(x,k)}(a_k)\right)\mathscr{F}^{z_{\varepsilon}(x,k)}(a_{k})}/{3^k}:a_k\in\left\{0,1,2\right\},x=\sum\limits_{k=-\infty}^{\infty}a_k/{3^k}\right\}}$$

which is the same as:

$$\scriptsize{f_{\varepsilon}(x)=\left\{\sum\limits_{k=-\infty}^{\infty}\Big(\text{sign}(\text{mod}(z_{\varepsilon}(x,k)+a_k,3))\cdot\text{mod}(z_{\varepsilon}(x,k)+a_k,3)\Big)/{3^k}:a_k\in\left\{0,1,2\right\},x=\sum\limits_{k=-\infty}^{\infty}a_k/{3^k}\right\}}$$

(i.e., $\mathscr{F}^{k}(a)=\text{mod}(a+k,3)$), such that for set $A\subseteq \mathbb{R}$ we want an $f:\mathbb{R}\to\mathbb{R}$ where:

\begin{equation} \small{\forall(\varepsilon_1>0)\exists(A\subseteq\mathbb{R})\forall(\varepsilon\in A)\exists(M>0)\forall(x\in\mathbb{R})\left(0<\varepsilon\le M\Rightarrow \left|f_{\varepsilon}(x)-f(x)\right|<\varepsilon_{1}\right)} \label{bjj}\tag{1} \end{equation}

Note: For $g_{\varepsilon}(x)$, we have $2\varepsilon$ (rather than $\varepsilon$) to avoid symmetry in the graph of $f$, and a $\text{sign}$ function in $f_{\varepsilon}(x)$ to allow negative outputs in the graph of $f$.

Attempt: It seems this function is:

1. Made of points "randomly" scattered across $\mathbb{R}\times\mathbb{R}$
2. Has infinite or undefined expected values (i.e., described in this paper, which is an extension of the Lebesgue density theorem; this paper which extends the Hausdorff measure using Hyperbolic Cantor sets; and this paper which applies a Henstock-Kurzweil type integral on a measure Metric Space).

For evidence, consider $f_{\varepsilon}(x)\mapsto f_{\varepsilon}(x,t)$ where:

$$\scriptsize{f_{\varepsilon}(x,t)=\left\{\sum\limits_{k=-\infty}^{t}\Big(\text{sign}(\text{mod}(z_{\varepsilon}(x,k)+a_k,3))\cdot \text{mod}(z_{\varepsilon}(x,k)+a_k,3)\Big)/{3^k}:a_k\in\left\{0,1,2\right\},x=\sum\limits_{k=-\infty}^{\infty}a_k/{3^k}\right\}}$$

and $x\in G_{r}:=\left\{c/d:c,d\in\mathbb{Z}, d\le r, -dr\le c\le dr\right\}$, such that:

$$\bigcup\limits_{r=1}^{\infty}G_r=\mathbb{Q}$$

where $x\in G_{10}$, we get the set of points $\left\{(x,f_{0.1}(x,10)):x\in G_{10}\right\}$, using Mathematica:

Clear["Global`*"]
ε = .1 ;(*We want ε->0*)
g[x_] := g[x] = 
  Piecewise[{{(2 ε)^(x/(2 ε)), 
     x < -2 ε}, {-1/x, -2 ε <= x < 0}, {0,
      x == 0}, {1/x, 
     0 < x < ε}, {ε^(-x/ε), \
ε <= x}}] (*g_ε(x)*)
z[x_, k_] := 
 z[x, k] = Round[g[x] k Sin[g[x] k]] (*z_ε(x,k)*)
f1[x_, k_] := 
 f1[x, k] = 
  RealDigits[N[x], 3, 2 k + 1, 
   k - 1](*Takes the digits of x from k's digit to the k-th digit*)
f[x_, t_] := 
 f[x, t] = 
  N[Sum[Sign[f1[x, t][[1]][[k]] + z[x, t - k + 1]] Mod[
      f1[x, t][[1]][[k]] + z[x, t - k + 1], 
      3]*3^(f1[x, t][[2]] - k + 1), {k, 1, 2 t + 1}]]
G[r_] := G[r] = 
  DeleteDuplicates[
   Flatten[Table[
     Range[-s*s, s*s]/s, {s, 1, r}]]] (*G_r: we want x=G_10 *)
ListPlot[Transpose[{G[10], 
   f @@@ Transpose@{G[10], ConstantArray[10, Length[G[10]]]}}]]

is the graph below

Question: Are there other special properties of $f$?