(* Function G of the partition of unity approach *)
FuncXi[s_,L1_,L2_] :=
Module [
{res, L, t},
(
If [s ≤  L1,
res =  1,
If [s < L2,
(
L = ( s - L1 ) / ( L2 - L1) ; 
t1 = 1 / (1 - L)^(1/2);
t2 = 1 / L^(1/2);
t =  ( Exp[ t1 - t2 ] )^2;
res = 1 / (1 +t ) 
),
res = 0
]
];

res
)
]

plotPeer = Plot[FuncXi[x,0.2,0.8] ,{x,0,1},PlotRange -> All, AxesLabel -> {"t", "Xi(t)"}]

rotplot = Plot3D[ FuncXi[ Sqrt[ x^2 + y^2], 0.2, 0.8], { x , -1 , 1 }, { y , -1 , 1 } , PlotPoints -> 80 , Mesh -> False, Boxed -> False, PlotRange -> Automatic]