Heaviside function

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).

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[GS]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1–5 , Acad. Press (1966–1968) (Translated from Russian) Zbl 0801.33020 Zbl 0699.33012 Zbl 0159.18301
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
[Sc]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951) MR2067351 MR0209834 MR0117544 MR0107812 MR0041345 MR0035918 MR0032815 MR0031106 MR0025615 Zbl 0962.46025 Zbl 0653.46037 Zbl 0399.46028 Zbl 0149.09501 Zbl 0085.09703 Zbl 0089.09801 Zbl 0089.09601 Zbl 0078.11003 Zbl 0042.11405 Zbl 0037.07301 Zbl 0039.33201 Zbl 0030.12601