Continuous time unit step and unit impulse functions

The Continuous Time Unit Step Function: The definition is analogous to its Discrete Time counterpart i.e.

u(t) = 0, t < 0
      = 1, t ≥ 0

The unit step function is discontinuous at the origin.

The Continuous Time Unit Impulse Function:  The unit impulse function also known as the Dirac Delta Function, was first defined by Dirac as

In the strict mathematical sense the impulse function is a rather delicate concept. The Impulse function is not an ordinary function. An ordinary function is defined at all values of t. The impulse function is 0 everywhere except at t = 0 where it is undefined. This difficulty is resolved by defining the function as a GENERALIZED FUNCTION. A generalized function is one which is defined by its effect on other functions instead of its value at every instant of time.