In other words can the acceleration change instantly? In direction and/or magnitude.
There are two aspects to this question.
In a problem, can you treat acceleration as changing instantly? (when in reality the change is actually finite but large)
Is there any way that acceleration can change instantly. Not as an approximation, but actually change instantly.
It seems the answer to 1) is yes. And the answer to 2) is no.
I can accelerate a block by pulling on a cord and at then just let go. The acceleration can be modeled as going to zero instantly. In this case, it seems that seems that the answer to 2) is also yes.
Velocity can never change instantly. That would mean that acceleration diverges and this is not physically possible. Position can't change instantly for obvious reasons.
This question has been explored here: https://www.reddit.com/r/askscience/comments/4eadfj/can_acceleration_change_instantaniously_or_is_it/