Heaviside step function

In mathematics, physics, and engineering the Heaviside step function is the following function,

H(x) = \begin{cases} 1 &\quad\hbox{if}\quad x > 0\\ \frac{1}{2} &\quad\hbox{if}\quad x = 0\\ 0 &\quad\hbox{if}\quad x < 0\\ \end{cases} $$ Note that a block function B&Delta; of width &Delta; and height 1/&Delta; can be given in terms of step functions (for positive &Delta;), namely

B_\Delta(x) = \begin{cases} \frac{ H(x+\Delta/2) - H(x-\Delta/2)}{\Delta} = \frac{0 - 0}{\Delta} = 0 & \quad\hbox{if}\quad x < -\Delta/2 \\ \frac{ H(x+\Delta/2) - H(x-\Delta/2)}{\Delta} = \frac{1 - 0}{\Delta} = \frac{1}{\Delta} & \quad\hbox{if}\quad -\Delta/2 < x < \Delta/2 \\ \frac{ H(x+\Delta/2) - H(x-\Delta/2)}{\Delta} = \frac{1 - 1}{\Delta} = 0 & \quad\hbox{if}\quad x > \Delta/2 \\ \end{cases} $$ The derivative of the step function is

H'(x) = \lim_{\Delta\rightarrow 0} \frac{H(x+\Delta/2) -H(x-\Delta/2)}{\Delta} = \lim_{\Delta\rightarrow 0} B_\Delta(x) =\delta(x), $$ where &delta;(x) is the Dirac delta function, which may be defined as the block function in the limit of zero width, see this article.