Heaviside Step Function - Zero Argument

Zero Argument

Since H is usually used in integration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of H(0). Indeed when H is considered as a distribution or an element of (see Lp space) it does not even make sense to talk of a value at zero, since such objects are only defined almost everywhere. If using some analytic approximation (as in the examples above) then often whatever happens to be the relevant limit at zero is used.

There exist, however, reasons for choosing a particular value.

• H(0) = ½ is often used since the graph then has rotational symmetry; put another way, H-½ is then an odd function. In this case the following relation with the sign function holds for all x:

• H(0) = 1 is used when H needs to be right-continuous. For instance cumulative distribution functions are usually taken to be right continuous, as are functions integrated against in Lebesgue–Stieltjes integration. In this case H is the indicator function of a closed semi-infinite interval:

• H(0) = 0 is used when H needs to be left-continuous. In this case H is an indicator function of an open semi-infinite interval: