[black_knight]

Very little of mathematics, like analysis? I am sure the analyst will care about all functions on the reals suddenly turning continuous. (Or rather losing the discontinuous ones)

Or what of commutative algebra and their beloved existence of maximal ideals!

[adgjlsfhk1]

you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.

[black_knight]

At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!