Heaviside function

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2020 Mathematics Subject Classification: Primary: 46F Secondary: 26A45 [MSN][ZBL]

The Heaviside function $H: \mathbb R \to \mathbb R$, called also Heaviside step function or simply step function, takes the value $0$ on the negative half-line $]-\infty, 0[$ and the value $1$ on the positive half line $]0, \infty[$. It is often of little importance how the function is defined in the origin $0$, however common choices are

  • $0$, making the function lower semicontinuous;
  • $1$, making the function upper semicontinuous;
  • $\frac{1}{2}$ compatibly with the convention, in use in the theory of functions of bounded variation, of defining the pointwise value of $H$ as the average of the right and left limit.

The Heaviside function is a function of bounded variation, in particular a jump function according to the terminology introduced by Lebesgue. General jump functions are indeed those functions of bounded variation $f$ which can be written as $f(x) := \sum_{i\in \mathbb N} c_i H (x-\alpha_i)$ for a suitable choice of the sequences $\{c_i\}$ and $\{\alpha_i\}$.

The generalized derivative (in the sense of distributions) of $H$ is the Dirac delta-function $\delta_0$, i.e. the measure which assigns to each set $E\subset \mathbb R$ the value \[ \delta_0 (E) = \left\{ \begin{array}{ll} 1 &\quad\mbox{if } 0\in E\\ 0 &\quad\mbox{if } 0\not\in E\, . \end{array} \right. \] (cp. also with Function of bounded variation).


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