The lemma states: Assume $f$ is bounded on $[a,b]$ and continuous on $[a,b]$ except at $a$. Then $f$ is integrable on $[a,b]$.
I am really stuck on this one. I would really appreciate some help. Thanks!
The lemma states: Assume $f$ is bounded on $[a,b]$ and continuous on $[a,b]$ except at $a$. Then $f$ is integrable on $[a,b]$.
I am really stuck on this one. I would really appreciate some help. Thanks!
Let $\epsilon > 0$ be given. Now, our problem is at $a$, so we could like to isolate $a$. Choose $x_0 = a$ and $x_1$ close enough to $a$ so that $(\max_{[x_0,x_1]}f(x)-\min_{[x_0,x_1]}f(x))(x_1 - x_0) < \frac{\epsilon}{2}.$ I'll let you fill in the details of why this is possible, but you must use boundedness of $f$. Now, $f$ is continuous on the interval $[x_1,b]$ hence integrable, and so we can find a partition $\{y_i\}$ of that interval so that $\sum_{i=1}^n (\max_{[y_i,y_{i+1}]}f(x)-\min_{[y_i,y_{i+1}]}f(x))(x_1 - x_0) < \frac{\epsilon}{2}.$ How can we combine this partition with $x_0,x_1$?