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Alsosaid1987 (talk | contribs) definition of function symbolically Tags: nowiki added Visual edit |
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==Properties==
The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], <math display="inline">c(x)</math> goes from 0 to 1 as <math display="inline>x</math> goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of exponent ''α'' = log 2/log 3) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.
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