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The world is both discrete and continuous at the same time.
Quantum fields are continuous, but our observations are discrete. This is impossible to understand without studying the basics of quantum mechanics.
According to current data, the universe is continuous, and our attitude to this does not matter at all. Of course, there is such a thing as the Planck length, which seems to be the minimum possible distance, because at smaller distances the very concept of space does not exist (any attempts to study it at high energies will end up with the formation of a black hole of Planck mass)
Then I quote the Wiki:
There is an assumption that there are spacetime cells with the minimum possible length equal to the fundamental length. According to this hypothesis, the degree of influence of space quantization on transmitted light depends on the size of the cell. The study requires intense radiation that has traveled as far as possible. Currently, a team of scientists has used data from the GRB 041219A gamma-ray burst survey, carried out with the European Integral Space Telescope. GRB 041219A is one of the brightest gamma-ray flares in the entire observation period, and its source is at least 300 million light-years away. Observation of the Integral made it possible to estimate the cell size several orders of magnitude more accurately than all previous experiments of this kind. Analysis of the data showed that if the space grain size exists at all, it should be at the level of 10-48 meters or less.
It's like arguing that light is a particle or a wave. Here, approximately, it is possible to give similar proofs of both the discreteness and continuity of the world (Universe).