So, what I think is that for some continuous output and any epsilon you care to name, one can construct a totally normal computer with resources 1/delta that can approximate the continuous output to within epsilon.

Proceeding from there, the more interesting question (and the most observable question) is more like the computational complexity question - does delta shrink faster or slower than epsilon? If it shrinks sufficiently faster for some class of continuous outputs, this means we can build a real-number based computer that goes faster than a classical computer with the same resources.

In this sense, quantum computers are already hypercomputers for being able to factor numbers efficiently, but they're not quite what I mean. So let me amend that to a slightly stronger sense where the machine actually can output something that would take infinite time to compute classically, we just only care to within precision epsilon :P