Editing Heaviside step function (section)

== Zero argument ==
Since {{mvar|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 {{math|''H''(0)}}. Indeed when {{mvar|H}} is considered as a [[distribution (mathematics)|distribution]] or an element of {{math|''L''{{isup|∞}}}} (see [[Lp space|{{math|''L{{isup|p}}''}} 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 [[#Analytic approximations|examples above]]) then often whatever happens to be the relevant limit at zero is used.

There exist various reasons for choosing a particular value. 
* {{math|''H''(0) {{=}} {{sfrac|1|2}}}} is often used since the [[graph of a function|graph]] then has rotational symmetry; put another way, {{math|''H'' − {{sfrac|1|2}}}} is then an [[odd function]]. In this case the following relation with the [[sign function]] holds for all {{mvar|x}}: <math display="block"> H(x) = \tfrac12(1 + \sgn x).</math>
Also, H(x) + H(-x) = 1 for all x.
* {{math|''H''(0) {{=}} 1}} is used when {{mvar|H}} needs to be [[right-continuous]]. For instance [[cumulative distribution function]]s are usually taken to be right continuous, as are functions integrated against in [[Lebesgue–Stieltjes integration]]. In this case {{mvar|H}} is the [[indicator function]] of a [[closed set|closed]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{[0,\infty)}(x).</math> The corresponding probability distribution is the [[degenerate distribution]].
* {{math|''H''(0) {{=}} 0}} is used when {{mvar|H}} needs to be [[left-continuous]]. In this case {{mvar|H}} is an indicator function of an [[open set|open]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{(0,\infty)}(x).</math>
* In functional-analysis contexts from optimization and game theory, it is often useful to define the Heaviside function as a [[Multivalued function|set-valued function]] to preserve the continuity of the limiting functions and ensure the existence of certain solutions. In these cases, the Heaviside function returns a whole interval of possible solutions, {{math|''H''(0) {{=}} [0,1]}}.