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This is a problem on which I've been working for some time, but that I can't prove

If $f$ is bounded and integrable on the improper sense on $[a,b]$, does it mean that it is Riemann-integrable on $[a,b]$?

My idea would be to use a theorem that states that if a function has a finite number of discontinuities and is bounded, then it is Riemann-integrable. But I'm not sure how to write it down rigourously.

DeepSea
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aga7689
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  • Excuse my ignorance, but what do you mean by "integrable on the improper sense on [a,b]"? – Eman Yalpsid Apr 17 '16 at 18:42
  • That for every [c,d] included in [a,b] minus a finite set of points C, the function is Riemann integrable and the limit to the points in C exists. Also, that the limite to the infinite bounds exists (assuming a = -oo or b = oo) – aga7689 Apr 17 '16 at 18:55
  • Without loss of generality it can be assumed that $f$ is bounded on $[a, b]$ and is Riemann integrable on $[a, c]$ for any $c \in [a, b)$ and $\lim_{x \to b^{-}}\int_{a}^{x}f(t),dt$ exist. The question then asks to prove that $f$ is Riemann integrable on $[a, b]$. see http://math.stackexchange.com/q/1301170/72031 and http://math.stackexchange.com/q/311395/72031 – Paramanand Singh Apr 19 '16 at 10:08

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