Finite realization of irrational transfer functions

There is at least one (rather trivial) way to formalize "implementability" that has the property that the only implementable functions are rational. Suppose that the implementation is defined as a continuous mapping $f:\mathbb R^{N+1}\to \mathbb R^N$ such that the last coordinate of the image of the current state concatenated with the next input produces the next output. This model makes a reasonable (IMHO) compromise between allowing the infinite computation precision and disallowing some clever encoding of the entire past into a single number. The proof of the rationality of the implementable functions is then very simple. Since there is no continuous injection from $\mathbb R^{N+1}$ to $\mathbb R^N$, there are two beginnings of length $N+1$ that result in the same state of the computer. If we continue with $0$ input after that, we will have equal outputs from there on, resulting in a finite recurrence relation for $h$.