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Can we introduce functions with a finite number of step discontinuities and still achieve uniqueness? Such functions are Riemann-integrable correct? So, does uniqueness hold for some classes of ##L^1## functions?

If not, can we identify the points where the trajectories are not unique, and decide "which" trajectory is chosen with some book-keeping of the inertia or "direction" of a trajectory?

If you think about this physically, a car or bike on a 2D surface might go in a loop-de-loop, it's trajectory will overlap with itself at multiple points. You'll notice this doesn't magically stop the car from attaining a smooth, continuous trajectory, even though mathematically, multiple trajectories may intersect with themselves or each other and and are therefore not unique.

Analogously, I don't believe uniqueness should truly be a problem for non-unique solutions to differential equations, so long as we can keep track of the "direction" or "inertia". But, how do we do that? What constraint or assumptions do we introduce to differential equations to attain continuous and practical trajectories that may model physical phenomena where trajectories are not necessarily unique?