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This work is concerned to study a kind of stability for mappings f : M → N, preserving the stratification defined by ψ : M → Q, which we call ψ-stability. We say that f,g : M → N are ψ-equivalents if there are diffeomorphisms h : M → M and k : N x Q → N x Q, which k(y,q) = (k₁(y,q),k₂(q)), such that (g,ψ) = k⁰(f,ψ)⁰h. And f is ψ-stable, if for all g sufficient near to f (in the Whitney's topology), follows that g and f are ψ-equivalents. If M ande N are compact and ψ: M → Q stable, the above definition is equivalent to the stability of the diagram M (f, ψ) > N x Q π > Q; also locally those concepts agree. The main objective is to characterize the ψ-stability when N is not compact, in particular to study R^P < ψ f > Rⁿ, M compact, which deep results on R < ψ M f > R, R < ψ M f > R² < ψ M f > R. Finally, local forms are given in the cases R, 0 < ψ Rⁿ, 0 f > R, 0 R²,0 < ψ R², 0 f > R, 0, R, 0 < ψ R², 0 f > R², 0 and R, 0 < f Rⁿ, 0 g > Rⁿ, 0 R, 0.