From t=0 onwards I changes slowly and V changes abruptly across the inductor.
At time t=0, the voltage across the inductor equalises the battery voltage; nevertheless, Lenz's Law states that this induced EMF will always be opposed to the polarity of the battery. The voltage across the inductor is equivalent to the voltage of a battery because the inductor at time zero behaves like a second battery of the same voltage linked in reverse.
Because current can never be zero, voltage across the inductor decreases with time. If it did, there would be no back EMF to stop the current from flowing through the inductor because the magnetic field would not be changing. As a result, the inductor will become less of an open circuit as the current increases over time. The inductor will essentially behave like a resistor.
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