fun[b_, x_] := 1/(2 b)*(Abs[x]^2 - Max[Abs[x] - b, 0]^2)
ParametricPlot[{u, fun[a, u]}, {u, -2, 2}, {a, 0, 1},
ColorFunction -> {ColorData["Rainbow"][#4] &}, PlotPoints -> 60]
Experts may be able to assist in removing artifact and definitely improve.