Editing Indicator function (section)

==Smoothness==
{{See also|Laplacian of the indicator}}
In general, the indicator function of a set is not smooth; it is continuous if and only if its [[support (math)|support]] is a [[connected component (topology)|connected component]]. In the [[algebraic geometry]] of [[finite fields]], however, every [[affine variety]] admits a ([[Zariski topology|Zariski]]) continuous indicator function.<ref>{{Cite book|title=Course in Arithmetic|last=Serre|pages=5}}</ref> Given a [[finite set]] of functions <math>f_\alpha \in \mathbb{F}_q\left[\ x_1, \ldots, x_n\right]</math> let <math>V = \bigl\{\ x \in \mathbb{F}_q^n : f_\alpha(x) = 0\ \bigr\}</math> be their vanishing locus. Then, the function <math display="inline">\mathbb{P}(x) = \prod\left(\ 1 - f_\alpha(x)^{q-1}\right)</math> acts as an indicator function for <math>V.</math> If <math>x \in V</math> then <math>\mathbb{P}(x) = 1,</math> otherwise, for some <math>f_\alpha,</math> we have <math>f_\alpha(x) \neq 0</math> which implies that <math>f_\alpha(x)^{q-1} = 1,</math> hence <math>\mathbb{P}(x) = 0.</math>

Although indicator functions are not smooth, they admit [[weak derivative]]s.  For example, consider [[Heaviside step function]] <math display="block">H(x) \equiv \operatorname\mathbb{I}\!\bigl(x > 0\bigr)</math>  The [[distributional derivative]] of the Heaviside step function is equal to the [[Dirac delta function]], i.e. <math display=block>\frac{\mathrm{d}H(x)}{\mathrm{d}x}= \delta(x)</math>
and similarly the distributional derivative of <math display="block">G(x) := \operatorname\mathbb{I}\!\bigl(x < 0\bigr)</math> is <math display=block>\frac{\mathrm{d}G(x)}{\mathrm{d}x} = -\delta(x).</math>

Thus the derivative of the Heaviside step function can be seen as the ''inward normal derivative'' at the ''boundary'' of the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain {{mvar|D}}. The surface of {{mvar|D}} will be denoted by {{mvar|S}}. Proceeding, it can be derived that the inward [[normal derivative]] of the indicator gives rise to a ''[[surface delta function]]'', which can be indicated by <math>\delta_S(\mathbf{x})</math>:
<math display=block>\delta_S(\mathbf{x}) = -\mathbf{n}_x \cdot \nabla_x \operatorname\mathbb{I}\!\bigl(\ \mathbf{x}\in D\ \bigr)\ </math>
where {{mvar|n}} is the outward [[Normal (geometry)|normal]] of the surface {{mvar|S}}. This 'surface delta function' has the following property:<ref>{{cite journal |last=Lange |first=Rutger-Jan |year=2012 |title=Potential theory, path integrals and the Laplacian of the indicator |journal=Journal of High Energy Physics |volume=2012 |issue=11 |pages=29–30 |arxiv=1302.0864 |bibcode=2012JHEP...11..032L |doi=10.1007/JHEP11(2012)032|s2cid=56188533 }}</ref>
<math display=block>-\int_{\R^n}f(\mathbf{x})\,\mathbf{n}_x\cdot\nabla_x  \operatorname\mathbb{I}\!\bigl(\ \mathbf{x}\in D\ \bigr) \; \operatorname{d}^{n}\mathbf{x} = \oint_{S}\,f(\mathbf{\beta}) \; \operatorname{d}^{n-1}\mathbf{\beta}.</math>

By setting the function {{mvar|f}} equal to one, it follows that the [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] integrates to the numerical value of the [[surface area]] {{mvar|S}}.