The question asks to whether the theorem "bounded function $f$ on $[a,b]$ that is continuous on $(a,b]$ is Riemann integrable" to explain whether these functions are Riemann integrable:
a) $\sin^2(\frac 1x)$
b) $\frac 1x\cdot\sin(\frac 1x)$
c) $\ln x$
I think the answer is Yes, No, No. For the first one, the function is clearly bounded and continuous on $(0,1)$. For the second one, I'm not entirely sure because the function is continuous on $(0,1]$ but is it bounded? The last one the function is unbounded.
Could someone confirm and explain to me more clearly? Much appreciated!