Integration

The classical, Riemann integral of the 19th century runs into difficulties with certain functions. For example:

To integrate the function x ^−1/2 from 0 to 1, the Riemann integral itself does not apply. One has to take a limit of Riemann integrals from ɛ to 1 as ɛ ↓ 0.

Also, the Riemann integral lacks some completeness. For example, if functions f_n on [0, 1] are continuous and |f_n (x)| ≤ 1 for all n and x, while f_n (x) converges for all x to some f(x), then the Riemann integrals ∫ 0 1 f n ( x ) d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076197/b707e6d7-4cb3-4b50-a24d-13dd813ac7b0/content/inequ4_86_1.tif"/> always converge, but the Riemann integral ∫ 0 1 f ( x ) d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076197/b707e6d7-4cb3-4b50-a24d-13dd813ac7b0/content/inequ4_86_2.tif"/> may not be defined.