In 1971 Fediĭ proved in [3] the remarkable theorem that the linear second order partial differential operator L f u ( x , y ) ≡ { ∂ ∂ x 2 + f ( x ) 2 ∂ ∂ y 2 } u ( x , y ) is hypoelliptic provided that f ∈ C ∞ ( R ) , f ( 0 ) = 0 and f is positive on ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, in [16] to infinitely degenerate elliptic divergence form equations ∇ tr A ( x , u ) ∇ u = ϕ ( x ) , x ∈ Ω ⊂ R n , where the nonnegative matrix A ( x , u ) has bounded measurable coefficients with trace roughly 1 and determinant comparable to f 2 , and where F = ln 1 f is essentially doubling. However, in the plane, these variants assumed additional geometric constraints on f, such as f ( r ) ≥ e − r − σ for some 0 < σ < 1 , something not required in Fediĭ's theorem. In this paper we in particular remove these additional geometric constraints in the plane for homogeneous equations, and only assume that f is positive away from the origin.