How could we know that changes are not discrete in reality?

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The OP is asking about discrete time and I would say we do not know for sure that time is continuous. Physicists are fond of differential equations, which assume continuous time. These equations fit reality pretty well, and this may prejudice our understanding.

Scientists and engineers familiar with digital control systems and simulations know that the world can just as well be modeled by difference equations with discrete time. Who can say this is not the true nature of reality?

Quantum mechanics offers yet another key to understanding: Time has uncertainty. The time at which an event occurs cannot be known with infinite precision. (Time and energy are conjugate variables, so the product of their uncertainties is a non-zero constant.) An exact time cannot be associated with any natural event. So it doesn’t make sense to say that time, and the changes associated with time, can be infinitely subdivided as we imagine that they can.
 
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At the core, this is a question whether time (or space) are quantized. Who knows? At a certain level (measurability) it begins to become more and more indistinguishable and irrelevant. I for one would prefer quantization: I am a bit afraid of the continuum, it is so intractable in some ways.
 
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