Riemann integrability is a property of a function on a closed interval that allows for the calculation of its definite integral. A function is considered Riemann integrable if its upper and lower Riemann sums converge to the same value as the partition of the interval approaches zero.
In order to show that a function is Riemann integrable, one must demonstrate that its upper and lower Riemann sums converge to the same value as the partition of the interval approaches zero. This can be done by showing that the function is bounded and discontinuous at a finite number of points, or by using the Riemann integral definition to evaluate the integral.
The Riemann integral definition states that a function f(x) on a closed interval [a, b] is integrable if and only if, for any given ε > 0, there exists a partition P of [a, b] such that for any refinement P' of P, the difference between the upper and lower Riemann sums of f over P' is less than ε.
Yes, a function can still be Riemann integrable on a closed interval even if it is not continuous. The only requirement for Riemann integrability is that the function is bounded and has a finite number of discontinuities on the interval.
Some common examples of functions that are not Riemann integrable include the Dirichlet function, the Thomae's function, and the Cantor function. These functions have infinite discontinuities on the interval and therefore do not meet the requirements for Riemann integrability.