In the numerical solution of initial value problems in ODEs, a state-of-the-art method will now deliver an approximation to the true solution at any point x in the interval of interest, [a, b], and not just at the adaptively selected discrete meshpoints. For Runge-Kutta methods, these off-mesh approximations are usually generated using a continuous extension (CRK) of an underlying discrete RK formula. In the development of optimal-order CRK methods, the focus has been on developing CRKs for explicit discrete RK formulas. In this investigation we will consider and discuss in detail the issues that arise when developing and implementing reliable, optimal order CRKs for implicit RK methods. In particular we will derive and justify suitable error control and step selection stratagies for these methods. The ability of these new methods to effectively approximate the solution of stiff IVPs and delay differential equations will be demonstrated.