Thank you for providing the code. The approach is neat and fully meets the requirements.

Here's a mod to Eric's approach to put circles at the right ends of the lines of the plot:

f2[x_] := E^Mod[x, 2, -2]
myplot = Plot[f2[x], {x, -6, 4}, ExclusionsStyle -> Dashed,
  PlotStyle -> 
   Directive[
    Arrowheads[{{0.01, 1, 
       Graphics[{White, EdgeForm[ColorData[101][1]], Disk[]}]}}]
    ]
  ] /. g : {___, Directive[___, _Arrowheads, ___], ___} :> (g /. Line -> Arrow)

It puts them at ALL the right end points, whether they deserve an open dot or not. You'll have to manually put some conditions in, if you want to restrict which segments get dots (e.g, the last one on the right should not get one if x were to go up to 4.5).

Or you could remove the last dot with the following:

myplot // With[{pos = Last@Position[#, _Arrow]}, 
   ReplacePart[#, pos -> (Extract[#, pos] /. Arrow -> Line)]
   ] &

I would suggest the option Exclusions, but it does not work with your style of defining functions. Do real Piecewise definitions instead, which are built for symbolic computation. As for the open circles, you can do them only manually, I suspect.