A computable function, formulated in the sense of mu recursion, can compute a for or do loop over some (possibly infinite) integer range.
I was wondering if a suitable generalization exists that allows the computation of a real-domain function $f$ where $f(x)$ can be calculated given the values of $f(y)$ for all real $y<x$. This seems natural to ask, given that it is relevant if you want to describe e.g. a real-world physical system as an algorithm.
First of all, despite claims by several people here, there are models of computation which support computation on topological spaces. The "countable vs. uncountable" distinction is a red herring, arising from being familiar only with notions of computation on discrete spaces. A good introductory text on the topic of computable analysis is Klaus Weihrauch's Comuptable analysis – an introduction.
Your idea is very closely related to the principle of open induction, which states: if an open subset $S \subseteq [0,1]$ satisfies $(\forall y \in [0,1].\, y < x \Rightarrow y \in S) \Rightarrow x \in S$ then $S = [0,1]$. In words: if an open $S$ has the property "if all numbers smaller than $x$ are in $S$ then so is $x$" then $S$ is the whole interval.
It is easy enough to speculate a functional version of open induction, which we could call open recursion (caveat – I am speaking off the top of my head here): given a map $$h : \Pi (x : [0,1]) \, (f : [0,x) \to \mathbb{R}) .\, \Sigma (\epsilon > 0) (g : [0, x + \epsilon) \to \mathbb{R}) . \, g{\restriction}_{[0,x)} = f$$ there is a unique map $\mathcal{O} h : [0,1] \to \mathbb{R}$ such that $ \mathcal{O} h (t) = h \, t \, ((\mathcal{O} h){\restriction}_{[0,t)}) $ for all $t : [0,1]$, where $f{\restriction}_{[0,t)}$ is the restriction of $f$ to the interval $[0,t)$.
The principle of open induction holds in classical mathematics (exercise), from which it follows that open recursion is valid as well (another exercise). That is, the notation $\mathcal{O} \, h$ is well-defined for any $h$.
Now, the interesting question is whether we can compute with open recursion. In An induction principle over real numbers explained how to compute with open induction. I am fairty sure that her method can be adapted to open recursion, but it would take more time to write up the details than I can invest in answering this question. I would be very interested to hear, however, if my hunch is correct.
AFAIK, there are no modelizations of physical systems with functions of infinitely many independent variables (which would be your generalization of complete induction, I guess).
Instead, models that predict the current value of a continuous signal based on previous "samples" (and some smoothness property) evoke ordinary or partial differential equations. These are solved by analytical techniques when possible using symbolic integration, for which we do have discrete algorithms, or by numerical methods, for which we also have discrete algorithms.