According to my understanding of SR, if I travel at 0.8c relative to a line of clocks, I should see the clocks in front of me going 3 times faster than my own, and those behind me going 3 times slower than my own (Doppler effect). OK, so what happens at my exact location? I reckon that as I look nearer and nearer to my origin, I would see a discontinuity between the forwards and backwards directions.
That's bad enough, but at my origin there is no light travel delay, so my local time is my proper time, and the speed of the clocks should then be in accordance with gamma (4/3) at 0.8c.
So, my question is: how do I reconcile the 3 contradictory speeds that I should observe for a clock at my origin? I think it should be the value given by gamma, but I can't explain the discontinuity resulting from the Doppler effect both forward and back in close proximity to the origin.