  We present a method to construct a continuous extension (otherwise known as
dense output) for a numerical routine in the special case of the numerical
solution being a scalar-valued function exhibiting rapid oscillations. Such
cases call for numerical routines that make use of the known global behaviour
of the solution, one example being methods using asymptotic expansions to
forecast the solution at each step of the independent variable. An example is
oscode, numerical routine which uses the Wentzel-Kramers-Brillouin (WKB)
approximation when the solution oscillates rapidly and otherwise behaves as a
Runge-Kutta (RK) solver. Polynomial interpolation is not suitable for producing
the solution at an arbitrary point mid-step, since efficient numerical methods
based on the WKB approximation will step through multiple oscillations in a
single step. Instead we construct the continuous solution by extending the
numerical quadrature used in computing a WKB approximation of the solution with
no additional evaluations of the differential equation or terms within, and
provide an error estimate on this dense output. Finally, we draw attention to
previous work on the continuous extension of Runge-Kutta formulae, and
construct an extension to a RK method based on Gauss--Lobatto quadrature nodes,
thus describing how to generate dense output from each of the methods
underlying oscode.
