We consider singularly perturbed Dirichlet problems which are, in the simplest nontrivial case, of the typeε2u″(x)=f(x,u(x)) for x∈[0,1],u(0)=u0,u(1)=u1. For small ε>0 we prove existence and local uniqueness of solutions u=uε, which are close to functions of the typeu¯ε(x)=u¯(x)+φ0(x/ε)+φ1((1−x)/ε) with f(x,u¯(x))=0 for x∈[0,1] and withφ0″(ξ)=f(0,u¯(0)+φ0(ξ)) for ξ∈[0,∞),u¯(0)+φ0(0)=u0,φ0(∞)=0,φ1″(ξ)=f(1,u¯(1)+φ1(ξ)) for ξ∈[0,∞),u¯(1)+φ1(0)=u1,φ1(∞)=0, and we show that ‖uε−u¯ε‖∞→0 for ε→0. We do not suppose monotonicity of the correctors φ0 and φ1. And, mainly, we do not suppose any regularity besides continuity of the functions f(⋅,u) and u¯. Hence, the functions u¯ε approximately satisfy the boundary value problem for ε≈0 in a very weak sense only, and one cannot expect that ‖uε−u¯ε‖∞=O(ε) for ε→0, in general. For the proofs we use an abstract result of implicit function theorem type which was designed for applications to spatially nonsmooth singularly perturbed boundary value problems with nonsmooth approximate solutions.