Aside on the Dirac Delta Function

  Plot[DiracDelta[t],{t,-3,3},
    PlotStyle->{RGBColor[0,1,0]}];

Mathematica does not like DiracDelta[0]...that should not surprise you. Let's have some fun:

  DiracDelta[.000000001]

  DiracDelta[0]

Can Mathematica recreate the properties of the Dirac Delta function?

  Integrate[DiracDelta[t],{t,-Infinity,Infinity}]

  
  Clear[f]
  Integrate[f[t] DiracDelta[t],{t,-Infinity,Infinity}]

Wait until you see this.

  Integrate[DiracDelta[t],t]

Is this our Heaviside Step Function? Let's see.

  Plot[UnitStep[t],{t,-3,3},
     PlotStyle->{RGBColor[0,1,0]}];

  Integrate[UnitStep[t],t]

Is this our ramp function?

  Plot[t UnitStep[t],{t,-3,3},
    PlotStyle->{RGBColor[0,1,0]}];

I'd say Mathematica handles this pretty well. Wolfram Research, Inc. must know what they are doing. They must also have some people who know their theory.

Up to Inverse Laplace Transform Examples