Summary
The right-hand side in delay differential equations, when considered as an abstract ODE, suffers from a lack of classical continuous differentiability if the delay depends on the state. The right-hand side is only continuously differentiable k times from the space of k+m times continuously differentiable functions into the space of m times continuously differentiable functions. One might expect that this may impede high-order convergence for numerical methods based on piecewise polynomial approximations, since even high-order piecewise polynomial are only continuous at their interval boundaries. We show that this is not the case, arriving at a convergence result similar to those for constant delays.
