Pensieve header: A demo of the basic $C^\infty$ Lego blocks.

σ[x_]:=(Piecewise[{{(E^(-1/x)), x>0}, {0, x≤0}}])
Plot[σ[x],{x,-1,1}]

(Subscript[β, ϵ_])[x_]:=σ[ϵ+x]σ[ϵ-x]
Plot[(Subscript[β, 1])[x],{x,-2,2}]

(Subscript[β, a_,ϵ_])[z_]:=(Subscript[β, (ϵ^2)])[((Norm[z-a])^2)]
Plot3D[(Subscript[β, 0,1])[{x,y}],{x,-2,2},{y,-2,2},PlotPoints->100,PlotRange->All, Exclusions->None]

f[x_,y_]:= If[(x^2)<y<3(x^3),(((Subscript[β, 2(x^2),(x^2)])[y])/((Subscript[β, 2(x^2),(x^2)])[2(x^2)])), 0];
Plot3D[f[x,y],{x,-0.1,1},{y,-0.1,1}, PlotPoints->100,PlotRange->All, Exclusions->None]