morphism said:Your function isn't defined at zero, but let's ignore that. Where does Wikipedia say that it's not integrable? It is in fact Riemann integrable. Its set of points of discontinuity has measure zero.
That's a sufficient condition, not a necessary condition. The necessary and sufficient condition is that the function be discontinuous on a set of (Lesbegue) measure zero. (This function satisfies that condition)mathman said:It is Lebesgue integrable, but not Riemann. For Riemann, you need continuity except at a countable set.
Nusc said:
That is NOT the "Dirichlet function" the OP was talking about. The particular Dirichlet function Wikipedia is referring to (There are several) is discontinuous everywhere.For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral.
HallsofIvy said:What Wikipedia says is
For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral.
That is NOT the "Dirichlet function" the OP was talking about. The particular Dirichlet function Wikipedia is referring to (There are several) is discontinuous everywhere.
I don't see what you're objecting to. Are you arguing that the given function isn't Riemann integrable?rbj said:but for x > 0, the given function is always less than the Dirichlet function (where not zero, the given function 1/q is less than 1). (it appears to be periodic with any period that is 1/q for integer q.) and we know that the Dirichlet function has Lebesque integral of zero and if this is Riemann integrable, i think the two integrals (over the same limits) has to be the same, no?
Riemann Integrability is a mathematical concept that describes the ability of a function to be integrated using the Riemann integral. This means that the function can be divided into smaller intervals and the area under the curve can be approximated using these intervals.
In order for a function to be Riemann Integrable, it must be bounded on a closed interval and have a finite number of discontinuities. Additionally, the upper and lower Riemann sums of the function must converge to the same value as the interval size approaches zero.
The Riemann integral is calculated by dividing the interval into smaller subintervals and approximating the area under the curve using these smaller intervals. This is known as the Riemann sum. As the size of the intervals approaches zero, the Riemann sum becomes more accurate and approaches the true value of the integral.
Yes, a function can be Riemann Integrable even if it is not continuous. However, the function must still be bounded and have only a finite number of discontinuities in order to be considered Riemann Integrable.
Riemann Integrability is an important concept in mathematics as it allows for the calculation of definite integrals, which have many practical applications in fields such as physics, engineering, and economics. It also serves as a building block for more advanced concepts in analysis and calculus.