Editing Dirichlet function (section)

== Integration properties ==
{{unordered list
| The Dirichlet function is not [[Riemann integral|Riemann-integrable]] on any segment of <math>\R</math> despite being bounded because the set of its discontinuity points is not [[negligible set|negligible]] (for the [[Lebesgue measure]]).
| The Dirichlet function has both an upper [[Darboux integral]] (namely, <math>b-a</math>) and a lower Darboux integral (0) over any bounded interval <math>[a,b]</math> — but they are not equal if <math> a < b</math>, so the Dirichlet function is not Darboux-integrable (and therefore not Riemann-integrable) over any nondegenerate interval.
| The Dirichlet function provides a counterexample showing that the [[monotone convergence theorem]] is not true in the context of the Riemann integral.
{{Math proof|drop=hidden|proof=Using an [[enumeration]] of the rational numbers between 0 and 1, we define the function {{math|{{var|f}}{{sub|{{var|n}}}}}} (for all nonnegative integer {{mvar|n}}) as the indicator function of the set of the first {{mvar|n}} terms of this sequence of rational numbers. The increasing sequence of functions {{math|{{var|f}}{{sub|{{var|n}}}}}} (which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable.}}
| The Dirichlet function is [[Lebesgue integral|Lebesgue-integrable]] on <math>\R</math> and its integral over <math>\R</math> is zero because it is zero except on the set of rational numbers which is negligible (for the Lebesgue measure).
}}