Q. a) Show that $f(x)=\sin(1/x)$ is integrable on any interval (you can define $f(0)$ to be anything). b) Compute $\int_{-1}^1\sin(1/x)dx.$ (Mind the discontinuity)
Let $r , \delta\in R^+$ for $r>\delta$. Then, $f$ is continuous at $[-r,-\delta] \cup [\delta,r].$ Therefore, it is Riemann integrable on this interval. The problem here is to show $f$ is Riemann integrable on $[-\delta, \delta].$ I found the accepted answer solving this problem. But, although I read several times, I am still struggling to understand the answer. Especially, why the difference between upper and lower sum is within $\varepsilon/3?$ Could you elaborate on this?