See Unifying continuous and discrete physics, Part 1.

I claim that the problem of unifying continuous and discrete physics ultimately reduces to the problem of summing geometrical progressions. See formulas for fractional iteration which is based on my work.

The sum of geometrical progression $\sum (1+\lambda+\lambda^2+\lambda^3+\ldots+\lambda^{n-1}) = \frac{1-\lambda^n}{1-\lambda}$, where $\lambda$ is not a root of unity. For $\lambda=1$ the summantion is just $n$.

Note the special cases of the sum of geometrical progression exactly correspond to the classification of fixed points in the complex plane. Consider that the sum of geometrical progression is what $D^2 f^n(z)$ primarily consists of. Since all higher derivatives are ultimately based on the first and second derivative, they all are primarily the summations of geometrical progressions. Thus $f^n(z)$ itself is built on the summations of geometrical progressions. For a complex treatment. For the detailed derivations of $D^m f^n(z)$ see tetration summary.

While the summations of geometrical progressions are discrete in their most general form, the hyperbolic equation results in $\frac{1-\lambda^n}{1-\lambda}$ which is continuous. Likewise the parabolic case where $\lambda=1$ and the summation is just $n$. As opposed to the summations where $n$ is a whole number, $n$ can take complex values in the hyperbolic and parabolic cases. This is one of the simplest examples of analytic continuation.

While the current progress is important, it is only proven in the complex plane. This gives us another toy universe to consider, but current research needs to be extended from the complex number system to invertible matrices of both finite and infinite dimensions. One major issue is that in higher dimensions the classification of fixed points is more difficult to catalog. Since fixed points and other invariant structures like limit cycles are at the core of iterated functions, more progress needs to occur in this area.