This question lies in the intersection of real analysis and logic, so I try to keep things rather basic.
First of all, logicians care about the following kind of formula:
Let $\varphi(n, x)$ be a formula with parameters $n$ (over the natural numbers) and $x$ (over the real numbers) such that $(\forall n\in \mathbb{N})(\exists \text{ at most one } x\in [0,1])\varphi(n, x)$.
This kind of formula has a nice connection to real analysis via the following function:
$$ f(x):= \begin{cases} \frac{1}{2^n} & \text{in case $n$ is the least number such that $\varphi(n, x)$}\\ 0 & \text{in case $\neg\varphi(m, x)$ for all natural numbers $m$} \end{cases} $$
The function $f:[0,1]\to \mathbb{R}$ has bounded variation (in the sense of Jordan). Hence, theorems of real analysis (like the Jordan decomposition theorem) establish properties of the aforementioned formulas $\varphi$.
Secondly, my question is whether the above extends to two or more dimensions.
Now let $\psi(n, x, y)$ be a formula with parameters $n$ (over the natural numbers) and $x, y$ (over the real numbers) such that $(\forall n\in \mathbb{N})(\exists \text{ at most one } x\in [0,1])(\forall y\in [0,1])\psi(n, x, y)$.
Now consider the following 'two-dimensional' version of $f$ above: $$ g(x, y):= \begin{cases} \frac{1}{2^n} & \text{in case $n$ is the least number such that $\psi(n, x, y)$}\\ 0 & \text{in case $\neg\psi(m, x, y)$ for all natural numbers $m$} \end{cases} $$
Does $g$ belong to a (nice/known) function class from analysis? If not, is there a modification of $g$ that does?
