[math-fun] Turning discrete games into continuous games ??

Are there any examples of an obviously discrete game -- e.g., tic-tac-toe, checkers, chess, go, etc. -- that can be converted into a continuous game ? The first goal might be to make time continuous, and a subsequent goal might be to make space continuous. Making time continuous is a problem, because how do the two sides "take their turn" ? But "taking turns" is really a discrete version of continually synchronizing time between two agents. In the case of chess, it is a discretized version of an obviously continuous game of war, so perhaps a continuous version might be made.