Heaviside Unit Step Function

The Heaviside Unit Step Function defines functions encountering ideal On/Off: (See plot.)

The Heaviside unit step function turns on a function at t0

The switch (change) at t0 is in fact an impulse, i.e., the Dirac delta function.

Dirac Delta Function

The Dirac Delta Function, also known as the unit impulse function, describes ideal short impulses:(See plot.)

The Dirac delta function works like a sampling gate at t0,

The effect of the sampling gate accumulated through the domain t is the unit step function.