We prove three analytic continuation criteria for functions defined on the whole real line, positive half-line and a finite interval. The extended functions belong to certain reproducing kernel holomorphic spaces. The distinctive feature of our theorems is that no smoothness properties of functions to be extended are assumed in the hypotheses. Incidentally, new orthogonality properties of the Hermite and Laguerre polynomials and necessary and sufficient conditions for square summability with geometric weight θ k of the coefficients in these polynomials are proved.