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Is there a function f: R->R, such that:

[tex] \forall x \in \mathbb{R}: f(x) \neq 0 \wedge \forall a,b \in \mathbb{R}: \int_a^b f(x) dx = 0 [/tex]

I made this problem myself so I don't know, wheather it is easy to see or not. The integral is the Lebesgue integral.

I would say, that there should be such a function, but I don't know how to define it. It has to be discontinuos everywhere, but defining it as something for rational numbers and something else for irrational doesn't help because rational numbers have measure zero.

Any suggestions?

[tex] \forall x \in \mathbb{R}: f(x) \neq 0 \wedge \forall a,b \in \mathbb{R}: \int_a^b f(x) dx = 0 [/tex]

I made this problem myself so I don't know, wheather it is easy to see or not. The integral is the Lebesgue integral.

I would say, that there should be such a function, but I don't know how to define it. It has to be discontinuos everywhere, but defining it as something for rational numbers and something else for irrational doesn't help because rational numbers have measure zero.

Any suggestions?

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