I want to know why is it that a voltage in a capacitor cant change instantaneously and why the current in an inductor cant change instantaneously either.

Also, a capacitor stores energy, like a battery, right? Why is it that it can't be used to power, say, and electric train or a car.

a capacitor charges because of the collection of electrons. The electron's can't move from the wire to the plate instantly, so the capacitor can't charge instantly either.

Similarly for inductors. Since electrons have to physically move along the wire, the current is a continuous function.

The closest you can come to instant change is to break a connection. If current is flowing, it can't stop in zero time, so you get a spark as the electrons try to keep moving. Since air has a finite dielectric constant, the current can flow for an instant.

Actually, a capacitor can be used to power a motor, but the required charge is enormous, so it will work briefly. The motor careth not whence the current cometh.