The Heaviside step function is the function $H:\sR\to \sR$ defined as \begin{eqnarray*} H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\ 1/2 & \mbox{when}\,\, x= 0,\\ 1 & \mbox{when}\,\, x> 0.\\ \end{array}. Here, there are many conventions for the value at $x=0$ . The motivation for setting $H(0)=1/2$ is that we can then write $H$ as a function of the signum function (see this page). In applications, such as the Laplace transform, where the Heaviside function is used extensively, the value of $H(0)$ is irrelevant. The Fourier transform of heaviside function is $$\mathcal{F}_0 H(t)=\frac{1}{2}\left(\delta(t)-\frac{i}{\pi t}\right)$$ where $\delta$ denotes the Dirac delta centered at $0$ . The function is named after Oliver Heaviside (1850-1925) [1]. However, the function was already used by Cauchy[2], who defined the function as $$ u(t) = \frac{1}{2}\big( 1 + t/\sqrt{t^2}\big)$$ and called it a coefficient limitateur [3].

Bibliography

1

The MacTutor History of Mathematics archive, Oliver Heaviside.

2

The MacTutor History of Mathematics archive, Augustin Louis Cauchy.

3

R.F. Hoskins, Generalised functions, Ellis Horwood Series: Mathematics and its applications, John Wiley & Sons, 1979.