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An almost zero function

Let f:[a, b] \rightarrow \mathbb{R} be a Riemann integrable function. If f(x)=0 forall rationals of the interval [a, b] then prove that \bigintsss_a^b f(x) \, {\rm d}x =0 .

Solution

Since f is Riemann integrable this means that the set of discontinuities has a zero measure. Wherever f is continuous , it’s gonna be zero due to the rationals being dense. Thus, f is almost everywhere zero. But then its Lebesgue integral is zero and so is the Riemann integral.

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