Editing Normal distribution (section)

=== Zero-variance limit ===
In the [[limit (mathematics)|limit]] when <math display=inline>\sigma^2</math> tends to zero, the probability density <math display=inline>f(x)</math> eventually tends to zero at any <math display=inline>x\ne \mu</math>, but grows without limit if <math display=inline>x = \mu</math>, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary [[function (mathematics)|function]] when {{tmath|1=\sigma^2 = 0}}.

However, one can define the normal distribution with zero variance as a [[generalized function]]; specifically, as a [[Dirac delta function]] {{tmath|\delta}} translated by the mean {{tmath|\mu}}, that is <math display=inline>f(x)=\delta(x-\mu).</math>
Its cumulative distribution function is then the [[Heaviside step function]] translated by the mean {{tmath|\mu}}, namely
<math display=block>F(x) =
\begin{cases}
  0 & \text{if }x < \mu \\
  1 & \text{if }x \geq \mu\,.
\end{cases}
</math>