The continuity of physical processes is not something that can be proved. It is an assumption that is included or not included among those on which a mathematical model of the physical processes is based. The suitability of any such model is based on the comparison between its predictions and the experimental results.

In standard physics, the space and time are continuous, and nothing else. All the physical quantities that are derived from length or angles or time are continuous (this includes quantities like mass and energy, which are derived from time). Besides those, there are no other physical quantities that are continuous.

The mathematical models that assume the continuity of space and time match extremely well any experiments and for now there exists no evidence whatsoever that space and time are not continuous.

Besides the standard physics, there have been a few attempts to create mathematical models of the physical processes where the space and time are discrete. Nevertheless, these mathematical models did not succeed to demonstrate any advantage over the standard models with continuous space and time.