Let $g: [0,1] \rightarrow \mathbb R$ and $g(x)=\frac{1}{q} $ if $x=\frac{p}{q}$, otherwise $0$.Prove that g is Riemann integrable.

Asked Dec 09 '19 at 21:59

Active Dec 09 '19 at 22:03

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Let $g: [0,1] \rightarrow \mathbb R$ and $g(x)=\frac{1}{q}$ if $x=\frac{p}{q} $ otherwise $0$.Prove that g is Riemann integrable.

Hint: Analyze the discontinuities of $g$. Prove that if $x$ is irrational and if $\frac{p_n}{q_n}$ is a sequence of rationals converging to $x$, then $q_n \rightarrow \infty$.

My idea: I observe that this function has countably many discontinuities. If I prove that If a function has countably many discontinuity then it is Riemann integrable. Is this direction is right?

Anyone suggest some idea from the hint?

edited Dec 09 '19 at 22:03

Bernard

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asked Dec 09 '19 at 21:59

User124356

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Another possible duplicate – RRL Dec 09 '19 at 22:13

Let $g: [0,1] \rightarrow \mathbb R$ and $g(x)=\frac{1}{q} $ if $x=\frac{p}{q}$, otherwise $0$.Prove that g is Riemann integrable.

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