“Darboux’s theorem(Darboux integral)”的概念、定义、翻译、参考文献-科学参考

Darboux’s theorem(Darboux integral)

Mathematics

单词	Darboux’s theorem(Darboux integral)
释义	Darboux’s theorem(Darboux integral) Mathematics Darboux’s integral is an alternative theory of integration which is equivalent to Riemann’s. Let f(x) be a bounded real-valued function on the interval [a,b]. Then both of the following are well defined $\int_{\underline{a}}^{b} f (x) d x = sup {\int_{a}^{b} φ (x) d x \| φ is a step function, φ \leq f} .$ $\int_{a}^{\bar{b}} f (x) d x = inf {\int_{a}^{b} φ (x) d x \| φ is a step function, φ \geq f} .$ These are respectively referred to as the lower and upper Riemann or Darboux integrals of f. Then f is said to be Darboux integrable if the lower and upper integrals are equal, and this is equivalent to being Riemann integrable. The Dirichlet function on [a,b] has lower integral 0 and upper integral b–a and so is not Darboux or Riemann integrable.
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