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==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[x_1,\ldots,x_n]</math> let <math>V = \left\{ x \in \mathbb{F}_q^n : f_\alpha(x) = 0 \right\}</math> be their vanishing locus. Then, the function <math display="inline">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>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>P(x) = 0</math>.
Although indicator functions are not smooth, they admit [[weak derivatives]]. For example, consider [[Heaviside step function]] <math display="block">H(x) := \mathbf{1}_{x > 0}</math> The [[distributional derivative]] of the Heaviside step function is equal to the [[Dirac delta function]], i.e. <math display=block>\frac{d H(x)}{dx}=\delta(x)</math>
and similarly the distributional derivative of <math display="block">
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 [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] gives rise to a 'surface delta function', which can be indicated by <math>\delta_S(\mathbf{x})</math>:
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