newtonian mechanics - Why can't impulse be instantaneous?

We know from 2nd law of motion that $$\vec{F} = \frac{d\vec{p}}{dt}.$$

Now, a rate of change can be instantaneous. So, rate of change of momentum is instantaneous. But I doubt how can there be change of momentum at an instant?

But when it comes to impulse, it always acts for a certain time, however small (infinitesimal) it might be. If rate of change of momentum is instantaneous and force is instantaneous, why can't the impulse be instantaneous? What is the cause?

Can anyone also tell how can there be change of momentum at an instant?

Answer

Consider a superball falling toward a hard tile floor. The ball will bounce rather nicely. How to model this collision?

If you look very closely at the bounce, a normal force from the floor starts acting on the ball when the bottom of the ball hits the floor. At this point in time, the top of the ball doesn't "know" that the bottom of the ball has hit the floor. That information propagates upward as a finite velocity pressure wave in the ball. The top of the ball keeps falling downward for a while. The ball gets compressed, rebounds, and flies back upwards. Some energy is lost as the ball is compressed and then relaxes back to its round shape.

One way to model this is to ignore all those details of the bounce. Since the duration is so very short, the bounce can be modeled as an instantaneous change in momentum determined by the ball's coefficient of restitution. This is a simplification that intentionally ignore the details of the bounce. While the maximum force is very high compared to the weight of the ball, the force is never infinite, and the change in momentum is not instantaneous.

The same applies to any application where one models forces as impulsive. It's just a model, a simplification that lets one ignore the details. The force is never truly infinite, so the change in momentum is never truly instantaneous.