The beginning of a movement and the continuity of time? ( How can movement begin without a first beginning instant ?) The question I am asking is not original , it is a very old problem dating back from antiquity. I would like to know how modern science has solved it; or, maybe, in  which fallacious reasoning  the question originates, from the point of view of modern physics. 
There must be someting wrong in this reasoning, since it produces a contradiction. Where is the mistake lying? 
(1) Suppose that a ball , at time t(0) is in equilibrium on a horizontal plane. 
(2) The direction of the plane is changed. 
(3) The ball begins to move. 
(4) Let's say that at time t(1) the ball is actually moving. 
(5) Between t(0) and t(1) there is some time at which the ball was already moving, let's say t(n); and between t(n) and t(0) there is some time at which the ball was already moving, let's say t(m)... so there is no first time at which the ball was moving. ( in the same way, I would say,  as there is no smallest real number strictly greater than 0 ). 
(6) but , if there is no first time, how could the ball begin to move? If there is no instant t such that the movement began at t, how can the change from immobility to movement happen? 
 A: The answer is two-tiered.
First, from the point of view of classical physics time is a continuous Real parameter so there need not be a single point which is the first point where the velocity is not-zero. We can even consider instantly changing the slope at time $t_{1/2}$. At any moment of time $t> t_{1/2}$ we have a non-zero velocity, yet there is no single point which is the first point with non-zero velocity. 
The change from immobility to mobility is affected by the causes operating from time $t_{1/2}$ even though their effect is infinitesimally later. There is no problem with that, it's just what the equations say.
The second tier is to wonder whether things are discrete. Consider quantum mechanics. It is common to believe that measuring distances below the Planck length is impossible. If so, it kind makes no-sense to speak of a continuous change here, since we can only measure that the ball has moved and attained a velocity after a Planck length at least. The result is that there possibly is a moment when the ball first gained velocity, which is precisely the moment when it - in each iteration of the experiment - reached Planck length.
The belief that Planck length (and time) are limitations stems from considerations of quantum gravity, which we don't really understand so they could be wrong. But my gut tells me that something like this is more likely. That space and time would be emergent quantized quantities in the full quantum gravity theory.
A: If you assume that the plane is instantaneously tilted at $t=0$, then your observation at the end of point 5 is spot on, there is no first time the ball is moving, precisely because there is no smallest number greater than 0. There is, however, a greatest time the ball was stationary, namely $t=0$, and this reasoning is general.
A: This is a variant of the 'Achilles and the Tortoise' problem. It assumes that whatever instant t of time is considered, there is always an earlier instant between t and zero at which some smaller degree of movement is present, and so it is not possible to specify a time which the motion actually starts. The answer, which is more to do with mathematics than physics, is yes it is always possible to subdivide the interval between t and zero without end, but the value of the intervals thus formed converges to zero.
