## Extract

Riemann was the first to consider the theory of integration of non-continuous functions. As is well known, his definition of the integral of a function between the limits a and b is as follows:— Divide the segment (*a, b*) into any finite number of intervals, each less, say, than a positive quantity, or norm *d*; take the product of each such interval by the value of the function at any point of that interval, and form the sum of all these products; if this sum has a limit, when *d* is indefinitely diminished which is independent of the mode of division into intervals, and of the choice of the points in those intervals at which the values of the function are considered, this limit is called the integral of the function from *a* to *b*. The most convenient mode, however, of defining a Riemann (that is an ordinary) integral of a function, is due to Darboux; it is based on the introduction of *upper and lower integrals* (intégrale par excès, par défaut: oberes, unteres Integral). The definitions of these are as follows:— It may be shown that, if the interval (*a, b*) be divided as before, and the sum of the products taken as before, but with this difference, that instead of the value of the function at an arbitrary point of the part, the upper (lower) limit of the values of the function in the part be taken and multiplied by the length of the corresponding part, these summations have, whatever he the type of function, each of them a definite limit, independent of the mode of division and the mode in which d approaches the value zero. This limit is called the upper (lower) integral of the function. In the special case in which these two limits agree, the common value is called *the integral the function*.

## Footnotes

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- Received April 23, 1904.

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