Question

why inductor offers high impedance to a.c. but a low impedance to d.c. ?

Answer 1

The voltage induced due to alternating current (change in current in time) opposes the emf of the circuit. Hence the impedance increases.  An inductor is just a resistor in a DC circuit.  But it becomes a resistance + inductance in an AC circuit.

More details.

Any real inductor can be thought of as a resistance R and an inductor L together, as any real piece of wire has some resistance in it.

An Inductor develops voltage across its ends, when there is a change in the current passing though it.  The voltage induced across the inductor is due to self inductance (L).  According to laws of electromagnetic induction, we get
 
     L  di / dt = v    (voltage is proportional to time rate of change of current)

Impedance due to the inductor is given by
   $X_L=\sqrt{R^2+\omega^2L^2}$

So, if there is a DC in the circuit, then ω = angular frequency of the current (and voltage) is zero. (votage and current are constant).   Hence impedance is  only R.

The frequency of AC power at our homes is 50 hz, that means ω = 2π * 50 rad/sec.   In that case,  Impedance is  

   $X_L=\sqrt{R^2+100^2\pi^2L^2}$

Hence in an AC circuit, the impedance is more.