There were some miscommunication in the previous post so I'm reposting.
Below is the theorem and the proof from my class. I get stuck when it mentions "Find $P',T', mesh \, P' < \delta$…" Also, where the heck did $|f(t_j' – f(t_j)| \Delta x > 2\epsilon$ come from?
Can someone explain step by step? Thank you so much!
Theorem: Let $f \in R[a,b]$. Then $f$ is bounded.
Proof: The integrability of $f$ implies $I = \int_a^b f(x)dx$. Let $\epsilon > 0$. Then there exists $\delta >0$ such that for any partition pair $P,T$ with $mesh \, P < \delta$, $|R(f,P,T)-I| < \epsilon$. Suppose $f$ unbounded. Find $P',T', mesh \, P' < \delta$ such that $|R(f,P',T')-I| \ge \epsilon$, which is possible since $f$ is unbounded. We know there exists interval $[x_i, x_{i+1}]$ where $f$ unbounded. Let $P'=P$. Try to change $T$. Pick $t_i' = t_i$ for $i \ne j$. But for $j = i$, let $t_j'$ be such that $|f(t_j') – f(t_j)| \Delta x > 2\epsilon$. Then $P', T'$ gives $|R(f,P',T')-I| > \epsilon$.
Best Answer
Suppose $f: [a,b] \rightarrow \mathbb{R}$ is unbounded.
$1$. Show that for any partition $P = \{a = x_0 < x_1 < \ldots < x_n = b\}$ of $[a,b]$ and any $M > 0$, one can choose sample points $x_i^* \in [x_i,x_{i+1}]$ such that
$|R(f,P,x_i^*)| = |\sum_{i=0}^{n-1} f(x_i^*)(x_{i+1}-x_i)| > M.$
(Hint: If $f$ is unbounded on $[a,b]$, it must be unbounded on at least one subinterval $[x_i,x_{i+1}]$. Choose one such subinterval; pick the sample points in every other subinterval arbitrarily, and then choose the sample point in the fixed unbounded subinterval last of all.)
$2$. Deduce that $f$ cannot be Riemann integrable on $[a,b]$.