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In classical mathematics, characteristic functions of sets only take values {{math|1}} (members) or {{math|0}} (non-members). In ''[[fuzzy set theory]]'', characteristic functions are generalized to take value in the real unit interval {{closed-closed|0, 1}}, or more generally, in some [[universal algebra|algebra]] or [[structure (mathematical logic)|structure]] (usually required to be at least a [[partially ordered set|poset]] or [[lattice (order)|lattice]]). Such generalized characteristic functions are more usually called [[membership function (mathematics)|membership function]]s, and the corresponding "sets" are called ''fuzzy'' sets. Fuzzy sets model the gradual change in the membership [[degree of truth|degree]] seen in many real-world [[predicate (mathematics)|predicate]]s like "tall", "warm", etc.
==Smoothness==
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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 [[algebraic geometIn the algebraic geometry]] of [[finite fields]], however, every [[affine variety]] admits a 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>.
A particular indicator function is the [[Heaviside step function]]
<math display="block">H(x) := \mathbf{1}_{x > 0}</math>
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