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Imagine a multistage rocket and the changes in acceleration.

Figure 4-3 in https://www.ibiblio.org/apollo/Documents/lvfea-AS506-Apollo1... shows this for Apollo 11.

I imagine if you zoom in far enough on those points you have the acceleration continuously changing as pressure slowly builds in the engines over several microseconds.

I was thinking more of the instant you shut off engines and disconnect 130,000 kg of mass of stage one.

There is an interesting Δa/Δt while fuel is consumed and mass changes.

There are discontinuities to the graph when engines are shut down and stages decoupled.

That’s the essence of a legitimate question: over small enough time periods (as the bolts explode over a non-zero period of time), is it continuous or discontinuous?

Over a macro scale, it’s discontinuous, of course.

It's nature, it's continuous at small enough scales.

But, checkout Zeno's paradox for more on your philosophical questions

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