Show that$f$ is in $R [0,1] $ (Riemann Integrable) and find find $\int_{0}^{1} $

Question

Show that $f$ is in $\mathscr R[0,1]$ (Riemann integrable) and find$\int_{0}^{1} f$

$f(x)=$ \begin{cases} 1 & 0 < x<1 \\ \\ 0 & x=1 ,x=0 \end{cases}

So I am having trouble with $U(P,f)$ and $L(P,f)$. Please help me as i need to study this.

would $P=[1,1-1/n,1]$ and would $\triangle x_i =1/n$

Answer 1

Fix $\epsilon > 0$. By the Archimedian property, you can choose $N \in \mathbb{N}$, so for $n > N$

\begin{align*} \frac{1}{n} < \frac{\epsilon}{2} \end{align*}

Then choose a partition $P = \{x_0=0, x_1= \frac{1}{n}, \dots, x_{n-1}= \frac{n-1}{n}, x_n=1\}$ and observe that

$\inf\{f([x_{i-1},x_i])\} = \begin{cases} 0 \quad \text{if } i=1 \ \mathrm{or \ } i=n \\ 1 \quad\text{otherwise} \end{cases}$

If you compute $L(P,f)$, you should have it equals $1-\frac{2}{n}$. Hence,

$U(P,f) - L(P,f) = 1 - (1-\frac{2}{n}) = \frac{2}{n} < \epsilon$