If a particle was at position X for zero time, was it there?

[hmmm27]

If he was only up on the horse for an instant of zero duration then he'll be on the ground soon enough.

That would be a point(s) contact for "Help my uncle get a leg over the horse".

 

[A.T.]

If the amount of time is zero, the speed of the particle is infinite.

Unless the distance traveled during that zero time is also zero. Then the speed is indeterminate, it can have any finite value. A position is just a point and has no size, so a point-like object travels zero distance, while being at that position.

 

[Vanadium 50]

I can't believe this is still going on.

You have a baseball traveling according to x = 2t + 1. At t = 3 you take a picture and sure enough, the photograph has a baseball at x = 7. Does anyone seriously want to argue that the baseball isn't "really" there? There's a picture and everything.

 

[hmmm27]

What I find interesting is that the OP's actual question - if he meant 2 0dim points - is non sequitur because - outside of a coordinate driven abstract, two 0dim points will never coincide : neither intersection nor duration. You need at least a total of 2 (spatial) dimensions between two objects to get an intersection, and at least 3 to get a duration of the intersection.