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Hello Helmut,
Thanks for your question, it is getting at core issues for physics. In my opinion the issue is the math-physics boundary
You are convinced, that velocity v = Delta_x/Delta_t is valid from v = 0 until v = oo without any exception or gaps.
This is entirely correct from the viewpoint of mathematics. Sure, make Delta_x =1 and Delta_t =0, the result will be infinity. Mathematics has no problem with this, however it is not reasonable physics. Why is it not reasonable physics? I will give you a challenge, find an event you can measure with a Delta_t = 0.
And sure you can find numbers that will make the velocity "smooth". But those numbers are mathematical and not obtained from physical measurements.
If you believe the math is the physics, you are stuck with your position (along with many members of this forum).
This is the problem I went after in "An Elephant in the Room". The equation for mass increase is "perfect" mathematically. Too bad the physical universe does not go along and shows it to be goofy. The remedy is easy once it is realized the equation has limits.
I hope this makes sense.
Let me know.
Thanks,
Don L.