What is the difference between the heaviside distribution and the dirac delta distriubtion?

Question

I know that the heaviside distribition is a piecewise function that deals with a discontinuous forcing functions but does the dirac delta function deal with the same type of situations? If so, what is the difference between them?

Answer 1

The Heaviside function is usually defined to be an antiderivative of the Dirac distribution:

$$ H'(x) = \delta(x) $$

For some $a < 0$ one gets:

$$ \int\limits_a^x \delta(\xi) \, d\xi = \left\{ \begin{array}{rc} 0 & x < 0 \\ 1 & x > 0 \end{array} \right. = H(x) $$