Why do we need continuous random variables since they can be approximated by discrete ones?

Question

I do not understand the motivation of developing the theory of continuous random variables. Given simple discrete random variables, the continuous ones can be well approximated.

Answer 1

Many formulas and concepts simplify in the continuous limit. Sampling with and without replacement become equivalent, complicated binomial coefficient sums become smooth Gaussians, continuous symmetry groups appear in the multivariate Gaussian distributions.

For the same reason there are real numbers and continuous functions used in calculus, although everything could be replaced by finite computations or sequences of those.