How to plot HeavisideTheta with y=0.5 at x=0, and similar piecewise functions

Something to get you started

Plot[HeavisideTheta[x], {x, -1, 1}, 
 ExclusionsStyle -> None,
 Epilog -> {
   {
    FaceForm[None],
    EdgeForm[ColorData[97][1]],
    Disk[{0, 1}, .025]
    },
   {
    ColorData[97][1],
    Disk[{0, .5}, .025]
    },
   {
    FaceForm[None],
    EdgeForm[ColorData[97][1]],
    Disk[{0, 0}, .025]
    }
   },
 Frame -> True,
 Axes -> False,
 ImageSize -> 500
 ]

I don't find the elegant way to un-clipped or un-filled the point,so I have to use White.

Clear[f];
f[x_] := Piecewise[{{1/2, x == 0}}, HeavisideTheta[x]];
Plot[f[x], {x, -5, 5}, 
 Epilog -> {Style[Point[{0, 1/2}], PointSize[Large], Blue], 
   Style[Point[{0, 1}], PointSize[Large], Green], 
   Style[Point[{0, 1}], PointSize[Medium], White], 
   Style[Point[{0, 0}], PointSize[Large], Cyan], 
   Style[Point[{0, 0}], PointSize[Medium], White]}, Axes -> False, 
 Frame -> True]