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I have been asked to find a sequence of numbers ($y_n$) such that $f(x)$ is not integrable on $[0, 1]$ where $f$ is defined as:

$$f(x)=\begin{cases}1 & \text{if }x\in(y_n)\\0 & \text{if }x\notin(y_n)\end{cases}$$ I have proven the function is integrable for certain sequences but am struggling to find a sequence that it is not integrable.

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    Riemann integrable .? – 2017-02-20
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    Every function defined like that is Lebesgue integrable. – 2017-02-20
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    Riemann integrable – 2017-02-20

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As $\mathbb{Q}$ is countable, consider $(y_n)_n$ a numeration of $\mathbb{Q}$. Then $f$ isn't integrable since $\mathbb{Q}$ is dense.

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    Thanks, this is what I was thinking but was not sure if that was considered a "sequence" or not. – 2017-02-20