The given paper deals with the construction and description of the integral funnel of all solutions of one-dimensional stochastic differential equations with unit diffusion. The notions of weak and strict regularity of the equations under consideration are proved to be equivalent and all the results known up to now on the existence and uniqueness of strict solutions follow from this. Specifically, it implies, that the weak solution with a distribution absolutely continuous with respect to the Wiener measure will automatically be strict. The solution of the innovation problem is shown to follow from this result in the case when the signal is a random variable.