What do you really see on a line of clocks as you pass by them at high speed? 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.
 A: I assume you used the formulae $f_o = fs\sqrt{\frac{1+v/c}{1-v/c}}$ for the clocks ahead of you and $f_o = fs\sqrt{\frac{1-v/c}{1+v/c}}$ for the clocks behind you. Those formulae do imply a singularity for the clock that is closest to you. Which equation to use?
The answer is neither. Those expressions assume the travel is along the line of sight to the source. There is a singularity because collisions are singularities. Your spaceship is plowing through the line of clocks. What you'll see in front of you are a series of clocks ticking faster than yours. Behind you, you'll see a cloud of pulverized clocks. Your spaceship had better have some very good forward shields. 
Your spacecraft presumably isn't doing that. Instead, you are flying parallel to the line of clocks, with some constant, non-zero distance between the spacecraft and the line of clocks. You need to use the more generic expession
$$f_o = fs \frac{\sqrt{1-\left(\frac v c\right)^2}} {1-\frac v c \cos \theta_o}$$
where $\theta_o$ is the angle between the clock in question and your line of travel, as observed by you. The sign convention here is that $\theta_o$ is positive for clocks in front of you, negative for clocks behind you.
The above expression reduces to the simpler expressions at the start of my answer for clocks very far in front of you and very far behind you. In between, you'll get a nice continuous change from faster in front, slower behind. The clock right next to you? It's a bit redshifted, and hence slower. Here $\cos \theta_0=0$, so in this case $fo=fs\sqrt{1-(v/c)^2}$. This is called the transverse Doppler redshift. This means that there's a clock just slightly ahead of you that is ticking at your own clock's rate.
A: The signal from the clock moving towards you is the Doppler shifted version of the value you "know" it to be - that is, first slow it down by gamma (clock moving relative to your frame of reference), then speed it up by Doppler shift.
Ditto, with sign reversed, for clocks behind you. Now the clock moving at right angles shows what you expect and there is no discontinuity.
A: If you consider a straight-on trajectory, there really will be a discontinuity. That is the same as with the audible Doppler effect.
There is a smooth drop in frequency of a fire truck's siren passing you on the street. The reason is, that there is a certain distance between you and the truck at the closest point. If that were not the case, i.e. the siren moves directly through you without the truck smashing you, you would perceive a sudden drop in frequency.
A: I am assuming we are talking about the one dimensional case in which we can "move through" clocks.
Assuming you synchronise the clocks in your own frame, those further away from you will show an older time than those closer because it takes longer for light to reach you from them.
On top of that, all clocks will be equally time dilated since they are all moving in the same frame.
The doppler effect enters because it will change the color of the clock - those ahead will appear blue and those behind red. The doppler effect does not change the actual time on the clock, though.
A: I don't have the reputation to comment, so I'll comment about David Hammen's (accepted) answer here.
His conclusion is correct, but mentions "Those formulae do imply a singularity for the clock that is closest to you" in reference to the forumale he thought you were referring to. He also mentions "In between, you'll get a nice continuous change from faster in front, slower behind" in reference to his latter formula involving a cosine. 
There is no singularity present in your original formulae that is not also present in David's. He has compared your equations (using a head-on speed) to his (using the proper, 'pythagorean' relative speed). 
In particular, if you are headed straight for a clock the angle $\theta_0$ will instantaneously flip $180^\circ$ when you pass it. At the same time $v$'s magnitude will flip. The 'singularity' is still there!
