I'm trying to understand the point addition and point doubling operations on elliptic curves for the purposes of elliptic curve cryptography. I've read this Wikipedia article for the formulae:
https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication
As well as this video for a derivation:
https://www.youtube.com/watch?v=XmygBPb7DPM
I now have a pretty good understanding of these operations for a regular elliptic curve. What confuses me, however, is how these formulae seem also to work when the curve is defined modulo some integer. Now the curve no longer looks like a smooth curve but rather a "cloud" of discrete points. We no longer have points on the "curve" for every $x$, but only for discrete integers. Moreover, the division operation (by $2p$) in point doubling is replaced by multiplication by the modular inverse.
So is there any kind of intuition why we can still apply the same formulae as in the continuous case? Why can the lambda still represent the derivative and why does the derivative even exist in this discontinuous case? In the continuous case, we can use these formulae and arrive at another point at the curve. In the modulo case, how do we even know that the new point satisfies the curve equation:
$$y^2 = x^3+ax+b$$
.. Now that we don't even find $y$ by taking the square root of the right hand side, but rather by finding its modular square root. All this seems very artificial, that is, we seem to simply replace all operators in the continuous case by their modulo "equivalents" and expect it to work.