[Animats]

The continuity of real numbers provides a clean theoretical basis for continuity of functions.

Exactly. Real numbers are an invention to make mathematics simpler and cleaner. If you don't have continuous reals, it takes extensive case analysis to prove relatively simple things for, say, a binary floating point representation. It's possible to do that; Boyer and Moore did work like that to formalize floating point and check its correctness. (That work was funded by AMD, after the famous Pentium divide bug.)

Newtonian mechanics assumed that the physical universe is described by real numbers. But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

Maybe reals should be viewed merely as a useful convenience for analysis, not some fundamental part of mathematics.

"God created the integers. All the rest is the work of Man." - Kronecker

[codethief]

> But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

This is actually not the case. Space and time are not regarded as quantized by the majority of physicists and there's also absolutely no evidence suggesting that. To the contrary, such quantization would lead to inconsistencies – e.g. with relativity – pretty quickly.

Fur further reading:

https://physics.stackexchange.com/questions/9076/does-quantu...

https://physics.stackexchange.com/questions/67899/is-time-qu...