Define $f$ on $[0,1]$ by
$f(x)=\begin{cases}x^2 ~~~~~~~~\text{if $x$ is rational}\\ x^3~~~~~~~~\text{if $x$ is irrational}\end{cases}$.
Then
- $f$ is not Riemann integrable on $[0,1]$.
- $f$ is Riemann integrable and $\int_{0}^{1}f(x)dx=\frac{1}{4}$,
- $f$ is Riemann integrable and $\int_{0}^{1}f(x)dx=\frac{1}{3}$,
- $\frac{1}{4}=\underline{\int_{0}^{1}}f(x)dx<\overline{\int_{0}^{1}} f(x)dx=\frac{1}{3}$, where $\underline{\int_{0}^{1}}f(x)dx$ and $\overline{\int_{0}^{1}} f(x)dx$ are lower and upper Riemann integral of $f$.
I am facing difficulty in solving the above problem, basically because I haven't solve this kind of question. I tried to see whether it is continuous or monotone, which implies R-integrable. But didn't make much headway in that direction. How to solve this kind of problems? Please give some hints or solutions. Thank you.
Best Answer
Take any partition $P$ of $[0,1]$ where $P=\{0<\frac{1}{n}<\frac{2}{n}<\ldots<\frac{n-1}{n}<\frac{n}{n}=1\}$
Since $x^2>x^3$ in each subinterval and $\Bbb Q$ is dense in $\Bbb R$ so $M_i=\dfrac{1}{n^2};m_i=\dfrac{1}{n^3}$
Then $U(P,f)=\sum_{i=1}^n M_i\Delta_{i}=\dfrac{1}{n}\{ \dfrac{1}{n^2}+\dfrac{4}{n^2}+\ldots\dfrac{n^2}{n^2}\}=\dfrac{n(n+1)(2n+1)}{6n^3}$ where $M_i=\sup f(x)$ in each subinterval
Again $L(P,f)=\sum_{i=1}^n m_i\Delta_{i}=\dfrac{1}{n}\{0+ \dfrac{1}{n^3}+\dfrac{8}{n^3}+\ldots \dfrac{(n-1)^3}{n^3}\}=\dfrac{n^2(n-1)^2}{4n^4}$ $m_i=\inf f(x)$ in each subinterval
Then as $n\to \infty$ ;$U(P,f)-L(P,f)=\dfrac{1}{3}-\dfrac{1}{4}=\dfrac{1}{12}\text{which does not go to } 0$
Hence $f$ is not $\mathcal R-$ integrable.
Also $\inf\{U(P,f)\}=\frac{1}{6}\{(1-\frac{1}{n})(2+\frac{1}{n}\}=\frac{1}{3}$
Similarly $\sup\{L(P,f)\}=\dfrac{1}{4}$