Editing Heaviside step function (section)

== Analytic approximations ==
Approximations to the Heaviside step function are of use in [[biochemistry]] and [[neuroscience]], where [[logistic function|logistic]] approximations of step functions (such as the [[Hill equation (biochemistry)|Hill]] and the [[Michaelis–Menten kinetics|Michaelis–Menten equations]]) may be used to approximate binary cellular switches in response to chemical signals.

[[File:Step function approximation.png|alt=A set of functions that successively approach the step function|thumb|500x500px|<math>\tfrac{1}{2} + \tfrac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}</math><br>approaches the step function as {{math|''k'' → ∞}}.]]
For a [[Smooth function|smooth]] approximation to the step function, one can use the [[logistic function]]
<math display="block">H(x) \approx \tfrac{1}{2} + \tfrac{1}{2}\tanh kx = \frac{1}{1+e^{-2kx}},</math>

where a larger {{mvar|k}} corresponds to a sharper transition at {{math|''x'' {{=}} 0}}. If we take {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, equality holds in the limit:
<math display="block">H(x)=\lim_{k \to \infty}\tfrac{1}{2}(1+\tanh kx)=\lim_{k \to \infty}\frac{1}{1+e^{-2kx}}.</math>

There are [[Sigmoid function#Examples|many other smooth, analytic approximations]] to the step function.<ref>{{MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function}}</ref> Among the possibilities are:
<math display="block">\begin{align}
 H(x) &= \lim_{k \to \infty} \left(\tfrac{1}{2} + \tfrac{1}{\pi}\arctan kx\right)\\
 H(x) &= \lim_{k \to \infty}\left(\tfrac{1}{2} + \tfrac12\operatorname{erf} kx\right)
\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.