We deal here with planar analytic systems x˙=X(x,ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by TΣ(q,ε) the time needed by a perturbed orbit that starts from q∈Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x˙=X(x,0) is a continuable critical orbit in a family of the form x˙=X(x,ε) if, for any q∈Γ and any Σ that passes through q, there exists qε∈Σ a critical point of TΣ(⋅,ε) such that qε→q as ε→0. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of x˙=X(x,0) is a continuable critical orbit in any family of the form x˙=X(x,ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x˙=X(x,0).