Differential Inequalities of Continuous Functions and Removing Singularities of Rado Type for J-holomorphic Maps

Among the results of Part Two there is the following: Proposition B. Let J be a C-smooth almost complex structure defined in R. Let v be a continuous map from the unit disc D ⊂ C into (R, J). Let u be either a constant map from D into R or a proper J-holomorphic and C2-smooth map from D into an open subset Ω of R. Assume that v is J-holomorphic near z if v(z) 6∈ u(D). If u ≡ u(0) then v is J-holomorphic on D; if u 6≡ u(0) and v(0) ∈ u(D), either v maps a neighborhood of 0 into u(D), or v is J-holomorphic near 0. If u ≡ u(0) the hypothesis is thereof that v is J-holomorphic at any point z such that v(z) 6= u(0). Proposition B generalizes the classical Rado Theorem.