To tell you the truth, at school I learned that Dirichlet function is actually what you call g(x):
> g(x) = 1 if x is rational
> 0 if x is irational
It exceptional property is that it is not continuous in every point. The function in the definition:
> f(x) 1/q when x=p/q
> 0 when x irational
is more sophisticated, and I saw it here for the first time ;).
But back to your question:
> could you explain the diference between these two functions
> (from the aspect of Riemann integrability)
So, the main difference, as I already said, is that f(x) is continuous in rational points, but g(x) is nowhere continuous. From the Riemann integrability, you wrote the right thing:
> f(x) is Riemann, integral = 0
> g(x) is not Riemann integrable
But from the Lebesque point of you they are both integrable, and integrals are equal to zero. So, on your question:
> wouldn't your ansewr apply to this function too?
> On an arbitrary interval [a,b] -> Lebesgue mesure is zero -> etc.
can I answer: yes! In principle, in the theory of Lebesque integral, one doesn't distinguish functions, which differ from each other on the set of measure zero, thus one frequently sees the words "almost everywhere". |
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