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 iterations of $\mathscr{F}$, we want to define a function with "pseudo-random" outputs in $\mathbb{N}$. If we define $g_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ where:

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

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

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

which contains "pseudo-random" outputs of $\mathbb{N}$, such that:

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

where for set $A\subseteq \mathbb{R}$, we want an $f:\mathbb{R}\to\mathbb{R}$ such that:

$$\forall(\varepsilon_1>0)\exists(A\subseteq\mathbb{R})\forall(\varepsilon\in A)\exists(M>0)\forall(x\in\mathbb{R})\left(0<\varepsilon\le M\Rightarrow \left|f_{\varepsilon}(x)-f(x)\right|<\varepsilon_{1}\right)$$


I'm not sure if this covers an "infinite expanse of space". (This would be interesting to graph).

I assume none of the expected values in the **preliminaries** give this function a unique and finite expected value. (Hopefully, someone can check.)