This is how I think of it too, and it makes some part of me want to believe that means I have an intuitive grasp do what’s going on. Except I don’t think that’s how math works. Rigor is necessary because an intuitive hand-wavey understanding is often wrong. I think there’s just as convincing a hand-wave available that dismisses the idea.
1) We have one point of the function.
2) We know one point of its derivative. We still don’t know any other points of the function.
3) We know one point of its second derivative. We still don’t know any other points of the function or its derivatives.
...
N) Same deal.
N+1) Same deal.
...
Inf) Now we know all the points in the function and also all the points in all the derivatives.
It looks absurd like this. Not to mention that we start with a countable set of points and derive a continuum from each.