The world is discrete, and computers are discrete. Cut out the middleman (continuous math).
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Possible. There's no rational reason to be so confident though.
It’s empirical. No one has ever made a continuous measurement. (E.g., light doesn’t get infinitely faint - it’s quantized.)
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Given that measurements involve bits of information, they have to be discrete. The map, however, is not the terrain. Position and energy of light may well be continuous, with only energy-exchange-via-bound-electron-orbits being quantized because of the properties of electrons.
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That’s an untestable proposition.
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