Yes, no such thing as continuity in the real world, real number are not physically real.

Buuut, they are very useful for making computations easiers.

I never disqualified the utility of the gaussian distribution, I simply pointed out the obvious that Gaussian != Uniform

> the expression "cryptographic hash function" itself implies apparent continuity of distribution.

No, it implies apparent uniformity of disitribution.

Continuity is another (topological) property.

f^(-1) (A) is an open set for every open set A.