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]

Z=NIntegrate[(Subscript[β, (1/2),(1/2)])[t],{t,0,1}];
θ[x_]:=(Z^(-1))NIntegrate[(Subscript[β, (1/2),(1/2)])[t],{t,0,x}]
Plot[θ[x],{x,-1,2}]