Why do inductors maintain the exact same current? As we know, a capacitor will maintain the  same voltage across its poles even if it is disconnected from a battery source. For example, in this problem:
http://www.unm.edu/~toolson/rc_circuit.gif
This is because the charge on the plates stays the same, and it is clear that the charge difference would maintain the same voltage.
So this brings me to my question. Switch the cap in that circuit for an inductor, and everything changes. As I have been told many times, now it is the current that stays the same. Indeed, almost counterintuitively, if an inductor is instantly disconnected from its power source and put into a series circuit with more resistance, the voltage across its terminals gets even bigger, even though there is more resistance.
So, I know that an inductor stores its energy in a magnetic field, and I am aware of the relation between the voltage across its poles and the change in current ($V=-L\frac{dI}{dt}$), but this does not tell me why the current should stay the same (at the very least, it isn't obvious).
It's clear that current would still continue to flow, but why is it the same current? Why does it maintain the status quo?
 A: The information about inductors you were given is not quite correct. Inductors resist a change in current flow, just like capacitors resist a change in voltage. 
When an inductor is switched into the circuit, the current starts to increase quickly, but the increasing magnetic field impedes the current. As the current increases, the magnetic field gets stronger.
When the inductor is disconnected from the circuit, the decrease in current allows the magnetic field to collapse. The decrease in the magnetic field forces electrons to move in the inductor. Since the electrons cannot go anywhere, a charge separation is created (increase EMF). The size of the inductor determines how long it will take for the magnetic field to dissipate.
A: 
Why does it maintain the status quo?

There is energy stored in the magnetic field, and the magnetic field is proportional to the current.
In order for the current to change, the magnetic field must change. 
By conservation of energy, that means the magnetic field energy must be transformed to some other kind of energy. Until you provide a mechanism for that to happen (for example a resistor that can turn the energy into heat), the current must remain constant.
A: Start with thinking of the case that you have lossless inductive loop in which the stationary (dc) current flows indefinitely, so then no loss and no current decay. Now assume that the sinusoid current is so large that the dissipation per cycle is insignificant when compared to the magnetic energy ("high Q") case, so that you have a constant amplitude oscillation over many cycles during which you wish to change the amplitude. But the change implies that you have to do something with the magnetic energy in the surrounding field. What would do that? In other words the current must go "somewhere". In RF amplifiers the anode (drain or collector) is connected through a coil (inductor) to the dc bias, the coil's inductance $L$ is large enough so that the its impedance $\omega L >> R_{loss}$, so that no matter what you do with the transistor (switch) the current is nearly constant through it over many, many cycles. It is not the same current for ever, it is just nearly the same for many cycles, and that is enough because the transistor changes its state much slower than the oscillations.
