f[x_]:=1/(1+x^2);

II[x_]:=HeavisideTheta[Abs[x]-1]*Sign[x]+HeavisideTheta[1-Abs[x]]*x;

Plot[{HeavisideTheta[1-Abs[x]]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{II[x]/2-II[x]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+0.3]/2-II[x-0.3]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+0.7]/2-II[x-0.7]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+1]/2-II[x-1]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+2]/2-II[x-2]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+3]/2-II[x-3]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+4]/2-II[x-4]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+5]/2-II[x-5]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+0.3]/2-II[x-0.3]/2,II[x+0.7]/2-II[x-0.7]/2,II[x+1]/2-II[x-1]/2,II[x+2]/2-II[x-2]/2,II[x+3]/2-II[x-3]/2,II[x+5]/2-II[x-5]/2},{x,-8,8},PlotRange->{0,1}]

Plot3D[II[x+y]/2-II[x-y]/2,{x,-8,8},{y,0,5},PlotPoints->60,Boxed->True,Axes->True]

II[x_]:=HeavisideTheta[Abs[x]-1]*Sign[x]/2+HeavisideTheta[1-Abs[x]]*(x-x*x*Sign[x]/2);

Plot[{II[x+0.3]/2-II[x-0.3]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+0.7]/2-II[x-0.7]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+1]/2-II[x-1]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+2]/2-II[x-2]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+3]/2-II[x-3]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+4]/2-II[x-4]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+5]/2-II[x-5]/2},{x,-8,8},PlotRange->{0,1}]

Plot[{II[x+0.3]/2-II[x-0.3]/2,II[x+0.7]/2-II[x-0.7]/2,II[x+1]/2-II[x-1]/2,II[x+2]/2-II[x-2]/2,II[x+3]/2-II[x-3]/2,II[x+5]/2-II[x-5]/2},{x,-8,8},PlotRange->{0,1}]

Plot3D[II[x+y]/2-II[x-y]/2,{x,-8,8},{y,0,5},PlotPoints->60,Boxed->True,Axes->True]

u[x_,t_]:=f[x+t]/2+f[x-t]/2+II[x+t]/2-II[x-t]/2;

Plot[{HeavisideTheta[1-Abs[x]]*(1-Abs[x])},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,0]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,0.3]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,0.7]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,1]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,2]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,3]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,4]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,5]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,6]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,7]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,8]},{x,-8,8},PlotRange->{0,1.2}]

Plot[{u[x,0],u[x,0.3],u[x,0.7],u[x,1],u[x,2],u[x,3],u[x,4],u[x,5],u[x,6]},{x,-8,8},PlotRange->{0,1.2}]

Plot3D[u[x,y],{x,-8,8},{y,0,5},PlotPoints->60,Mesh->None,Boxed->True,Axes->True]