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Numerade Educator

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Problem 71 Hard Difficulty

Let $ f(x) = 0 $ if $ x $ is any rational number and $ f(x) = 1 $ if $ x $ is any irrational number. Show that
$ f $ is not integrable on $ [0, 1] $.

Answer

$f$ is not integrable on [0,1].

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Video Transcript

We know that the interval I over and comma I plus one over and will always have a rational and irrational number regardless of how large end becomes as it continues on. Therefore, we know we have the limit As one approaches infinity. We know this is one and then we know the same thing. But when we have zero times one over end a zero. As you can see, the limit depends on the choice. Therefore, off is not possible to be integrated. It's not Intercable.