A solution of this equation is the spreading Gaussian, ρ ρ t ( x ) = 1 2 π π t e − − x 2 2 t , and, since the integral of ρ t is constant while the width is becoming narrow at small times, this function approaches a delta function at t =0, lim t → → 0 ρ ρ t ( x ) = δ δ ( x ) again only in the sense of distributions, so that lim t → → 0 ∫ ∫ x f ( x ) ρ ρ t ( x ) = f ( 0 ) for any smooth test function f.