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Describe continuity of dirichlet function in relevance with integration in detail
describe continuity of dirichlet function in relevance with integration in detail
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Krishna![best-answer]()
Let\u00A0\uFFFC\u00A0and\u00A0\uFFFC\u00A0be\u00A0real numbers\u00A0(usually taken as\u00A0\uFFFC\u00A0and\u00A0\uFFFC). The Dirichlet function is defined by\n\n\uFFFC\n\n(1)\n\nand is\u00A0discontinuous\u00A0everywhere. The Dirichlet function can be written analytically as\n\n\uFFFC\n\n(2)\n\n\uFFFC\n\nBecause the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version, sometimes itself known as the Dirichlet function (Bruckner\u00A0et al.\u00A02008), Thomae function (Beanland\u00A0et al.\u00A02009), or small Riemann function (Ballone 2010, p.\u00A011), can be defined as\n\n\uFFFC\n\n(3)\n\n(Dixon 1991), illustrated above. This function is continuous at irrational\u00A0\uFFFC\u00A0and\u00A0discontinuous\u00A0at rational\u00A0\uFFFC\u00A0(although a small interval around an irrational point\u00A0\uFFFC\u00A0contains infinitely many rational points, these rationals will have very large denominators). When viewed from a corner along the line\u00A0\uFFFC\u00A0in normal perspective, a\u00A0quadrant\u00A0of\u00A0Euclid's orchard\u00A0turns into the modified Dirichlet function (Gosper).