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Give an example of a positive function f on $[0,1]$ such that $f\in R([0,1])$ but $1/f \notin R([0,1])$ Thanks for your help

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What is $\,R([0,1])\,$? –  DonAntonio Dec 10 '12 at 4:29
    
riemann integrable –  Miguel Mora Luna Dec 10 '12 at 4:30
    
Then what's the Riemann-Stieltjes integral to do here? –  DonAntonio Dec 10 '12 at 4:31

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up vote 1 down vote accepted

$$f(x)=\begin{cases}\left(x-\frac{1}{2}\right)^2&x\neq\frac{1}{2}\\8&x=\frac{1}{2}\end{cases}$$

The above is Riemann integrable in $\,[0,1]\,$ , but $\,\displaystyle{\frac{1}{f(x)}}\,$ is not (why?)

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I edited my first answer since at first I didn't notice the positiveness requirement –  DonAntonio Dec 10 '12 at 4:36

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