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

The Heaviside unit step function turns on a function at

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

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