# An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable.

I have been given the example $$f_n(x)= \begin{cases} 1, & \text{if}\; x\in\{(x_k)_{k=1}^{n}\}, \\ 0, & \text{otherwise.} \end{cases}$$ Here the sequence $$(x_k)_{k=1}^{\infty}$$ is the sequence that "enumerates" elements of $$\mathbb{Q}$$. The sequence of functions converges pointwise to $$f(x)= \begin{cases} 1, & x\in\mathbb{Q}, \\ 0, & x\notin\mathbb{Q}, \end{cases}$$ which isn't Riemann integrable. However, I am not quite convinced that $$f_n$$ is integrable. I would like some help in seeing how $$f_n$$ is Riemann integrable. Also, I would like more examples of sequences of Riemann integrable functions $$f_n$$ that converge pointwise to a function $$f$$ that is not Riemann integrable.

• With $f_n(x)=1$ for $x=x_1,\ldots, x_n$, and $f_n(x)=0$ otherwise, each $f_n$ is continuous at all but finitely many points; thus, integrable. Dec 18, 2013 at 20:14
• Integrable means "Riemann Integrable"? Dec 18, 2013 at 20:15
• I see. So the integral of $(f_n)$ is 0? Dec 18, 2013 at 20:15
• Yes, I always forget. Sorry, I'll correct it Dec 18, 2013 at 20:15
• In your definition of $f_n$, I believe you should have $n$ as the upper bound on the sequence.
– user61527
Dec 18, 2013 at 20:17

For the $f_n(x)$, it comes down to the fact that the Riemann sum $1dx \rightarrow 0$ when the partition width is small-enough. Then you do this finitely-many times, and you get a total of $0$ for any $f_n(x)$. You can show that this result is true no matter how small you make the partition width $||P||$, i.e., you can show $|\Sigma f(x)dx-0 |<\epsilon$ for any partition width.
• I am not sure whether making the partition width small enough would suffice since $\mathbb{Q}$ is dense in $\mathbb{R}$. Could you elaborate? Dec 18, 2013 at 20:24
• But remember that $f_n(x)$ is the integral where only $x_1,x_2,..,x_n$ are non-zero. So you don't need to worry about all terms in $\mathbb Q$ , nor density. Dec 18, 2013 at 20:26