Sign into an existing account

https://www.youtube.com/watch?v=Fb2ei6lD-d8

I had heard of the Dirichlet Function: zero at irrational points and one at rational points. It is an example of Riemann's definition of the integral not working. But I didn't know why anyone cared.

This video sketches out a sequence of Riemann integrable functions that converge to the Dirichlet Function. And makes the point: we would really like a definition of the integral that make the limit of a sequence of integrals agree with the integral of the limit of the sequence of functions.

The video doesn't get into the technical details of how the Lebesgue integral does this, but it makes it clear what motivated Henri_Lebesgue

https://youtu.be/tSQ6JaWyM2Q?si=PTMFUr4lsZps3_wP

@indyreno2933 says, "In case everyone should know, recent studies render Elephas maximus polyphyletic, asiatic elephants are defined by the genus Elephas as a whole and the genus existed solely in Asia, which means the genus never had any species native to Africa.

Elephas maximus is polyphyletic because the extant asiatic elephants are now six separate species: the Indian Elephant (Elephas indicus), the Sri Lanka Elephant (Elephas maximus), the Bangladeshi Elephant (Elephas bangladeshiensis), the Indochinese Elephant (Elephas vietnamensis), the Sumatran Elephant (Elephas sumatrensis), and the Bornean Elephant (Elephas borneensis), these six species are not closely related to each other and are considered closer to many other extinct species.

The bornean elephant is more closely related to the extinct beyer's elephant (Elephas beyeri) and sulawesi dwarf elephant (Elephas celebensis) the clade containing the bornean elephant (Elephas borneensis), the †beyer's elephant (Elephas beyeri), and †sulawesi dwarf elephant (Elephas celebensis) is the sister group to the clade containing the †javan elephant (Elephas sondaicus), †blora elephant (Elephas hysudrindicus), and †ngandong dwarf elephant (Elephas indonesicus).

The sumatran elephant (Elephas sumatrensis) is basal to both clades and more derived than the indochinese elephant (Elephas vietnamensis), the clade containing the †chinese elephant (Elephas rubridens), †hainan elephant (Elephas hainanus), and †formosan elephant (Elephas taivanus) is most closely related to the clade containing all of the eight species of asiatic elephant in Southeast Asia.

The bangladeshi elephant (Elephas bangladeshiensis) is basal to both clades, the indian elephant (Elephas indicus) and sri lanka elephant (Elephas maximus) are both more closely related to both the †falconer's elephant (Elephas hysudricus) and the flat-headed elephant (Elephas platycephalus), the †persian elephant (Elephas persicus) appears as more derived than other asiatic elephants in western asia, but is basal to other asiatic elephants.

The †syrian elephant (Elephas asurus) has evolved before both the persian elephant and all of the asiatic elephants native further east. The clade containing the †turkish elephant (Elephas turcicus) and †caucasian elephant (Elephas caucasicus) is the sister to the clade containing all other asiatic elephants, which leaves the †jordan elephant (Elephas jordanensis) as the most basal of the twenty-two species of asiatic elephant and the †arabian elephant (Elephas arabicus) as the second most basal.