Instantaneously changing currents in RL circuit
We are talking about the mathematical models used to model a lumped-element network. 

In this context any current can change instantaneously, unless there is a reason it can't. The reason why \$i_L\$ can't change instantaneously is that that current flows in an inductor. 

Current flowing in an inductor cannot change instantaneously. Any attempt to force such a current to change instantaneously would generate a voltage spike of infinite amplitude across the inductor (a Dirac's delta pulse, from a mathematical POV).

In practice, trying to change a current in an inductor abruptly will lead to large voltage spikes (not infinite) until some nonlinear phenomenon will arise in the circuit (e.g. arcing) and will quench the spike.