Most people think in terms of mutation, not in terms of the abstract functions that cause values to change over time.
When I see a ball falling I don't think of it moving forward and accelerating in an infinite number of infinitesimal mutations---I see it as being in the state of falling and recognize the space of actions I could perform in time with it. There's mutation in the sense that I cannot arbitrarily go back in time. There's immutability in the sense that the rules governing that motion are not "staged" in any way outside of the interference of my hand.
A better example is perhaps catching a fly ball as an outfielder in baseball. I absolutely cannot be processing that motion as a set of mutations in order to function---instead, I predict according to a "mathematical pattern" if not a formula, where it will land based on scant observation of segments of its space-time trajectory.
And that's all more or less instinctual, I believe. It must be in order for me as an outfielder to execute my task. That stuff needs to just be built in to my brain.
So while ultimately FRP and physics might be "implemented" (ha, ha, philosophers forgive me) in discrete steps, there's a vital model of each which is continuous.
If you really want to get in the thick of it, take Conway's game of life (or NKOS if you can stomache wolfram): there is quickly in a point in a physical process where you don't know what happens next without the previous state.