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
example of a positive function-riemann stieltjes integral
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real-analysis
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0Then what's the Riemann-Stieltjes integral to do here? – 2012-12-10
1 Answers
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$f(x)=\begin{cases}\left(x-\frac{1}{2}\right)^2&x\neq\frac{1}{2}\\8&x=\frac{1}{2}\end{cases}$
The above is Riemann integrable in $\,[0,1]\,$ , but $\,\displaystyle{\frac{1}{f(x)}}\,$ is not (why?)
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1I edited my first answer since at first I didn't notice the positiveness requirement – 2012-12-10