I think of discrete dynamical systems
$$x_{t+1}-x_{t}=\Delta x_t= f(x_{t})$$
and continuous dynamical systems
$$\frac {\partial x_t}{\partial t} = f(x_t)$$
as totally separate formalisms. However, sometimes I want to be agnostic about whether a dynamical system I'm considering is discrete or continuous. Is there a formalism that has both of these as special cases? (similar to how the Lebesgue integral has the discrete sum and the continuous (e.g. Riemann) integral as special cases)
bonus question: It would be even better if the formalism had partial differential equations and higher order differential equations also as a special case.