Consider a simple model:
A single photon of light in a vacuum travels from (-1,0) to (0,0) , where it hits a reflective point at the origin at time T.
CASE 1: Suppose the velocity of light follows a continuous curve (i.e. the plot of v VS t for this photon is a continuous curve). Then, at t < T it was traveling at v=+c, and at t > T it is traveling at v=-c, so under continuity assumptions, the velocity must be 0 at some point, and so the speed must be 0 at some point, not possible.
CASE 2: Otherwise, suppose the plot of v VS t is a discrete curve (i.e. the v vs t graph is simply +c when t<T and -c when t>T). Then the photon goes from a velocity of +c to -c instantaneously at time T. Notwithstanding that its exact velocity would be incomputable at the exact time of contact (is it +c or is it -c at time T?), we know that this implies instantaneous reflection. Hence, discrete curve implies instantaneous reflection.
Thus, the contrapositive of this conclusion is that in a world with non-instantaneous reflection, we cannot have discrete curve, i.e. we must be in CASE 1. But in the real world where absorption and emission (i.e. reflection) takes time, we arrive in CASE 1, and so the mere fact that light changes directions seems to be contradictive to the fact that an individual photon of light must remain at the same speed all the time. (I understand that the energy transfer takes time and hence slows down light, but consider again how its velocity could ever go from +c to -c in the real world (i.e. non-instantaneous reflection) as that would imply it taking on a speed of 0 in between, in this continuous-curve case).
What am I misunderstanding here? It appears that it can't be the same photon that is being emitted, correct? That would actually settle the contradiction here, though it's not the accepted paradigm.