We consider Burgers solution u = u(x,t,v); v > 0, satisfying a linear growth with respect to x. We associate with u the stochastic characteristic  . Here B is the Brownian motion. We show that if (x,t) is a continuity point of the inviscid Burgers solution u(x,t,0) then the stochastic characteristic converges uniformly in L² to the characteristic segment  .Our study contains the case where the initial value is singular