Domain-continuous visibility determination algorithms have proved to be very efficient at reducing noise otherwise prevalent in stochastic sampling. Even though they come with an increased overhead in terms of geometrical tests and visibility information management, their analytical nature provides such a rich integral that the pay-off is often worth it. This paper presents a time-continuous, primary visibility algorithm for motion blur aimed at ray tracing. Two novel intersection tests are derived and implemented. The first is for ray versusmoving triangle and the second for ray versusmoving AABB intersection. A novel take on shading is presented as well, where the time continuum of visible geometry is adaptively point-sampled. Static geometry is handled using supplemental stochastic rays in order to reduce spatial aliasing. Finally, a prototype ray tracer with a full time-continuous traversal kernel is presented in detail. The results are based on a variety of test scenarios and show that even though our time-continuous algorithm has limitations, it outperforms multi-jittered quasi-Monte Carlo ray tracing in terms of image quality at equal rendering time, within wide sampling rate ranges.