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asdf60
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the modified dirichlet function (1/q if x is rational = p/q, 0 if x is irrational) is integrable. How is it integrable? What is the upper step function?
Integrability of Modified Dirichlet Function & Upper Step Function
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In summary, the modified Dirichlet function is integrable and even Riemann integrable. This is because for any given value of m, there are a finite number of x values where f(x) is less than 1/m. By defining step functions at these points as 1 and the rest as 1/m, the step function integral can be made arbitrarily small and the infimum goes to 0. This means that there are infinitely many values of x where f(x) is nonzero.
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Hurkyl
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That one's even Riemann integrable, if memory serves.
Start with this question: for how many values of x is f(x) nonzero? (Infinitely many, I know that.
Be more specific!)
Start with this question: for how many values of x is f(x) nonzero? (Infinitely many, I know that.
Be more specific!)- #3
asdf60
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Yes, it is reimann integrable. Well after a bit of thinking, i figured it out. The trick is that for any m, there are finite number of Xs such that f(x) < 1/m. So define the step functions at those points as 1, and the rest as 1/m. But you can make the step functions width arbitrarily small at the points f(x) > 1/m, so they don't contribute to the step function integral. Then obviously, the infemum goes to 0.
Related to Integrability of Modified Dirichlet Function & Upper Step Function
1. What is the definition of integrability of a function?
The integrability of a function refers to the property of a function being able to be integrated, or finding the area under its curve. This means that the function has a finite value when integrated over a particular interval.
2. How is the integrability of a function determined?
The integrability of a function is determined by evaluating the Riemann integral of the function over a given interval. If the Riemann integral exists, then the function is considered to be integrable over that interval.
3. What is the difference between the Modified Dirichlet Function and the Upper Step Function?
The Modified Dirichlet Function and the Upper Step Function are both examples of non-integrable functions. However, the Modified Dirichlet Function is discontinuous everywhere, while the Upper Step Function is only discontinuous at a single point.
4. Why is the Modified Dirichlet Function not integrable?
The Modified Dirichlet Function is not integrable because it is discontinuous at every point in its domain. This means that the Riemann integral cannot be evaluated, as it requires the function to be continuous over the interval of integration.
5. Can the Upper Step Function be made integrable?
No, the Upper Step Function cannot be made integrable. Despite being discontinuous at only one point, it still does not meet the necessary requirements for integrability, which include being bounded and having a finite number of discontinuities.
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