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→Analytic approximations: Adding a non-analitic example |
→Zero argument: Added a property for step function when its value for zero argument is 0.5 Tags: Mobile edit Mobile web edit |
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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>
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