[adamnemecek]

Here's the thing, your description of the continuous set is already discretized. If we say an interval 4-6 we have captured the continuous space using only two numbers.

I know that this is a silly argument in some sense but this is something that you do naturally that you don't even think about it.

Do you see what I'm getting at? You capture extrema in the discrete space and the interpolate in the smooth space to recreate the smooth curve.