Newton's axioms and collision Newton's axioms for point particles states that the velocity of a point particle is differentiable. However when two object collide there is a jump in their respective velocities. So is "ideal" collision something that is outside Newton's axioms or there is an extended concept of differentiability that functions with a jump counts as differentiable?
 A: What you are describing as an "ideal" scenario is really an over-simplified one.
In reality, the objects colliding are not point particles.  They have volume.
Upon collision, the materials deform, causing them to slow down as they collide and press against each other.
In a perfectly elastic collision, this deformation would take all the kinetic energy from the objects, and store it internally in the deformations of the object.  Because it behaves elastically, as soon as the kinetic energy is gone, the objects will keep trying to force themselves back apart.  When we assume the energy goes fully back into kinetic energy, we see the velocities reversed.  It is essentially two springs touching.
This is not actually a discontinuity.  That part of the velocity/time curve would actually be a bit rounded, not a sharp point as is often shown in the analysis.
The only time you would get the discontinuity is for infinitely stiff elastic collisions, where they do not deform at all but still transfer the force back.  That is non-physical, so we only consider that when dealing with hypotheticals/idealized scenarios. 
