Each decimal number is equivalent to a function $f(x)$ in this number system. $f(x)$ is defined on the whole number line and only takes single digit integer values 0-9 everywhere. The conversion from function to decimal is given by:
$$\int_{-\infty}^{\infty} f(x)10^{x}dx$$
Does a unique function $f(x)$ exist in this number system (or maybe 'function system') for each decimal number?
On
The Heaviside step function $H(x)$ takes the value $0$ for $x\le 0$ and $1$ for $x>0$. We can use the Heaviside function to construct
$f_a(x)=H(a-x)$
which is $1$ if $x<a$ and $0$ everywhere else. Then we have
$\int^{\infty}_{-\infty}f_a(x)10^xdx=\int^{a}_{-\infty}10^xdx=\frac{1}{\ln 10}10^a$
If $b=\frac{1}{\ln 10}10^a$ then $a(b)=\frac{1}{\ln 10}(\ln \ln 10 + \ln b)$ and
$\int^{\infty}_{-\infty}f_{a(b)}(x)10^xdx=b$
This works for $b>0$ although it is not a very "natural" construction, and I am sure there are many alternatives.
The function $f$ is not uniquely defined.
For instance, the number $1$ can be represented by
$$f(x):=\begin{cases}0\le x\le \log_{10}(\log(10)+1)\to1\\\text{else }\to0\end{cases}$$
or by
$$f(x):=\begin{cases}0\le x\le \log_{10}\left(\frac{\log(10)}2+1\right)\to2\\\text{else }\to0\end{cases}$$
and in uncountably many other ways.
The continuum of $\mathbb R$ gives you way too many degrees of freedom for the representation to be unique.