Continuous is smooth and smoothing is averaging.
Discrete is separate, usually by a specified granularity size (binary) or kernel, which is a fancy pixel that can be of different shapes. [3×3, 5×5, 7×7 instead of 1×1 or 1x1x1 or 1x1x1x1, etc] — but usually you pick only one kernel and go with it for a consistent grid/matrix per operation.
Time, I won’t get into just yet.
When calculating continuous things, at some point when dealing with ground truths you have to choose a kernel / round it off / pick a granularity / choose a scale/zoom.
So, if you round to the 100th place, that’s the size of your grain. The grain is what stays in the pan and does not fall through the sieve. Yes, it’s like panning for gold.
The smoothness of continuous is, I believe, a function of our brain’s compression of information. I believe the Universe is indeed bumpy.
However, I do not believe it is bumpy in an evenly granular way.
Is it ultimately smooth when you go as deep as possible? I don’t think so. Smooth gives us a flexibility to adjust grain size / kernel / scale/zoom as needed but at a cost of complexity and difficulty to calculate as it is manipulating improbable things: smoothings of smoothings of smoothings of smoothings all down to the infinitesimal, a hypothetical smallest grain – the notion of grain and function of grain while leaving the choice ultimately up to the practitioner of the mathematical art.
So, my fascination of fractions and fractals and fractures, the gaps and cracks of things, I also know that it’s possible to pretend “as if” a surface is smooth _and_ solid _and_, a 0 or 1 — “it isn’t or it is” because for the purpose I am using the information for, the fraction/fracture/fractal/imperfectness of things is less important and is a distraction from purpose.
Fractures go down to electromagnetic fields level but practically speaking one can usually stay at the molecular level and work with classic physics and with chemistry, which is more pragmatic.
Yet, I love the stuff “at the bottom”. The larger grains can be boring at times — unless dealing with them en masse as if they are marching ants with independent thought but with limited decision-making choices.
So, I am a discrete guy but I’ll be quick to point out that there’s always a power of 10 smaller you can go. I want to believe in the continuous but it’s a smoothing/averaging operation.