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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!

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$\frac{1}{x} \sin(1/x)$ is not bounded: consider the sequence of points $x_n=\frac{1}{\pi/2+2n\pi}$ and notice $\sin(1/x_n)=1$.

Also, in all of these you should be a bit careful: you need to give some alternate definition of them at $x=0$ in order for the question of Riemann integrability to make sense. As your theorem shows, the choice of this value is of no consequence, but still, the concept of proper Riemann integrability is restricted to functions whose domain is a closed bounded interval.

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  • $\begingroup$ oh I see! Thanks so much! $\endgroup$
    – Rainroad
    Mar 21 '16 at 11:38

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