Let $f:[0,1]\rightarrow \mathbb{R}$ be a function. If $P=\{0=a_0 < a_1 < \cdots < a_n=1\}$ is any partition of $[0,1]$, then we define $U(p,f)$ and $(L(p,f)$, the upper and lower Riemann sums.
Suppose, we have the partition $P_n=\{0,\frac{1}{n}, \frac{2}{n},\cdots,\frac{n}{n}=1\}$ for $[0,1]$ and suppose we prove that $|U(P_n,f)-L(P_n,f)|$ tends to $0$ as $n$ tends to $\infty$. Can we conclude that $f$ is integrable on $[0,1]$?
I am considering this type of partition, for example, to check integrability of Thomae's function. (There was proof of the integrability but I was trying to pass through only specific types of partitions of $[0,1]$ and conclude the integrability; but I do not know, if this type of partition is sufficient for checking integrability.)
Best Answer
Yes, we can conclude that $f$ is Riemann integrable, according to an integrability criterion quoted in the accepted answer here.