Solve the below:
a) Justify the formula δ(2x) = 1/2δ(x) by limits and by duality;
b) Find a similar formula for δ(ax) when a > 0 and when a < 0;
c) We know that xδ(x) = 0; where δ is the delta function. On the other hand by Leibniz rule (xδ(x))'= δ(x) + xδ'(x) is apparently not zero. How can this paradox be explained?