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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.
== Non-Analytic approximations ==
Approximations to the Heaviside step function could be made through [[Non-analytic_smooth_function#Smooth_transition_functions|Smooth transition function]] like <math> 1 \leq m \to \infty </math>:
<math display="block">\begin{align}f(x) &= \begin{cases}
{\displaystyle
\frac{1}{2}\left(1+\tanh\left(m\frac{2x}{1-x^2}\right)\right)}, & |x| < 1 \\
\\
1, & x \geq 1 \\
0, & x \leq -1
\end{cases}\end{align}</math>
==Integral representations==
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