Let D be a bounded domain of the complex n-space Cn(n≥2), or more generally a pair (M,D) a finite manifold (cf. Definition 2.1), and we assume the boundary ∂D is a smooth and connected submanifold. It is well known by Hartogs-Osgood’s theorem that every holomorphic function on a neighbourhood of ∂D can be continued holomorphically to D. Generalizing the above theorem we shall prove that if a differentiable function on ∂D satisfies certain conditions which are satisfied for the trace of a holomorphic function on a neighbourhood of ∂D, then it can be continued holomorphically to D (Theorem 2-5). The above conditions will be called the tangential Cauchy Riemann equations.