No one has mentioned the crucial point. These kinematic quantities
pertain to the motion of something abstract, usually a point in space
defined by a formal procedure defining the center of mass of some
physical object. Alternative meanings are certainly possible, such as
the instantaneous velocity of a particular point identified in some
system by a particle, say an oxygen atom in a turbulent river. What
has been implicit here is that one is talking about the motion of a
massive extended object - and idealizing it by discussing the motion
of a point having no size - or substantial reality!
The discussion of whether acceleration can change discontinuously in
time is not a physical discussion; it is at best metaphysical, and
more properly it is simply an empty question because it uses the word
"can" in a physical rather than a mathematical sense, while the
question is purely mathematical. Sure the acceleration can change
discontinuously. Nothing more is needed than the hypothetical
statement that it does, e.g. "Given that the acceleration of an
automobile starting from rest is 0.1 g for the first ten seconds and
then -0.2 g for the next fifteen seconds, what is the position..."
Until an oscilloscope capable of splitting time on the Planck scale
is offered by Tektronix, the accurately measured acceleration of no
physical object will be found to be discontinuous. Discontinuous
acceleration is readily achieved, however, with the pen.