I want to know the rigorous mathematical definition of displacement, which is applicable to physics ?

All we are taught is that its a vector quantity as opposed to a scalar such as distance.

And I have a velocity time graph, which again, I am not too sure on a few aspects of it.

For example, if a ball is dropped and bounces back up, and this repeats until the ball stops, each time the velocity decreases, the graph shows a straight sloped line going down to -3, then a staight line with same gradient from +2.

But what about the velocity in between 2 and -3 ?

and what meaning can be given to a vertical line on the velocity time graph ?

thanks

Roger