What's the difference between the Riemann & Darboux Integrals?

[swampwiz]

I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.

 

[fresh_42]

https://en.wikipedia.org/wiki/Riemann_integral

Darboux worked with lower and upper sums, Riemann with a mean value. There is no essential difference, as e.g. to Lebesgue integrals.

 

[mathman]

Riemann integrals and Darboux integrals have different definitions. However they are equivalent.

 

[Periwinkle]

One of the two authors, Frederic Riesz, describes the question in detail in points 10 and 13 of Chapter I of the book by F. Riesz and B. Sz. Nagy, Leçons d'analyse functionelle.

There are two types of Darboux-integrals: upper and lower. These integrals belong to two interval functions

$$(\beta-\alpha) \sup_{\alpha \leq x \leq \beta} f(x), ~~~~~ (\beta-\alpha) \inf_{\alpha \leq x \leq \beta} f(x). $$
If the first interval function can be integrated, then we are talking about the upper Darboux-integral, if the second, then the lower Darboux-integral. Maybe both exist, perhaps just one, possibly none.

In cases where both exist and are even equal, it is said that the function can be integrated in the Riemann sense. So Darboux's integrals have much more functions than Riemann's integrals.