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→Formulation: Added the piece-wise definition also to the other two kinds of Heaviside function |
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\end{align}</math>
These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)|distributions]]. In general, however, [[pointwise convergence]] need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem|convergence holds in the sense of distributions too]].)
In general, any [[cumulative distribution function]] of a [[continuous distribution|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function|cumulative distribution functions]] of common probability distributions: the [[logistic distribution|logistic]], [[Cauchy distribution|Cauchy]] and [[normal distribution|normal]] distributions, respectively.
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\end{align}</math>
where the second representation is easy to deduce from the first, given that the step function is real and thus is its own [[complex conjugate]].
== Zero argument ==
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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]].
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