Why can't the current in a solenoid be changed rapidly? Why can't the current in a solenoid be changed instananeously? My reasoning is since the current in a solenoid is given by a integral:
$$I(t) = \frac{1}{L} \int_{t_0} ^{t_f} V_Ldt$$
then if $t_f = t_i $, the integral is zero, and there can be no current. But what happens to the solenoid itself if the current is changed instantaneously? And how do we define how slowly we need to change the current? Is there a fixed time-scale?
 A: The current changes instantaneously, but as a continuous function, that is: not suddenly from zero to a finite value, without passing for all intermediate values.
The magnitude of $V$ in the solenoid is a funtion of how fast the current changes. And the bigger the inductance, the bigger $V$ for the same rate of change of current.
It is the same differential equation for $F = m\frac{dv}{dt}$. When we push an object without any friction, it starts to move immediately. For huge accelerations it is necessary huge forces. And the greater the mass, the greater the force for the same acceleration.
A: I'm not sure about your background -- which physics level you got. So I will only use "school" arguments to explain the effect.
The key characteristic of a coil with a running current is that it is an electro-magnet. However, changing a magnetic field $B$ is something which takes time. This is due to Lenz's law, which says something like
if we try to change a magnetic field, the magnetic field induces a current
such that the induces current generates a B field, which opposes the B change 

So, let's start with $B=0T$. Now, if we suddenly apply a voltage to a solenoid, such that the current would generate a $B$ field in the positive $z$ direction,    the induced current counteracts its rapid/sudden change by flowing such that the induced $B$ pointing in the negative $z$ direction. Hence, the real and induces current flow in the opposite direction. The sum of these two generates the well known exponential grow of the current in the coil.
A: If you answer the question of how the magnetic field of a coil is created, you have an understandable model for such processes.
The point is that you have to consider the electron with its magnetic dipole. The existence of the magnetic dipole is the intrinsic (existing under any circumstances) property of electrons. Remember that in permanent magnets, the alignment of these dipoles is the reason for the macroscopic magnetic field.
The equation (or relationship) for the occurrence of a magnetic field due to a current channelled in a wound wire is empirical (found by observation and repeated measurements many times), and deeper explanations are needed to see how it happens.
The reason for the magnetic field of a current in a coiled wire is the same as for permanent magnets: the common alignment of the electrons magnetic dipoles. The key point is that this alignment costs energy. That is why for a wire of the same length as a coiled wire an alternating current flows with lower resistance and with less energy losses.
At any time the current in a coil changes, the directions of the magnetic dipoles of the electrons change and this takes time. So turning off the current causes the dipoles of the electrons to return to their original position, which moves the electrons some more and the current is supported for a while.
If this is not imaginable for you at once, please remember (or read) about the Lorentz force respectively the Hall effect. The external magnetic field interact with the moving electrons and the electrons get deflected.
Lenz's rule is thus the expression for the behaviour of moving electrons with their magnetic dipoles under the influence of a macroscopic magnetic field.
