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I've been trying to solve this but with no luck, and couldn't find anything online. The problem is:

Let $f(x)$ be a riemann integrable function over the interval $[a,b]$. Show that the function $e^{f(x)}$ is integrable over the interval $[a,b]$.

Thank you!!

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  • $\begingroup$ By integrable, do you mean Riemann integrable or Lebesgue integrable? $\endgroup$ Jun 30, 2020 at 9:01
  • $\begingroup$ Riemann integrable $\endgroup$
    – Orokusaki
    Jun 30, 2020 at 9:50

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If you are talking about Lebesgue integration this is false. $-ln x$ is integrable on $(0,1)$ but $e^{-\ln x}$ is not integrable.

If you are talking about Riemann integration this follows from the fact that a bounded function is Riemann integrable iff it is continuous almost everywhere.

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  • $\begingroup$ I understand what you are saying, but the question is about a closed interval [a,b] where in your example -lnx is integrable on an open interval (0,1). Is it still ok to assume that? $\endgroup$
    – Orokusaki
    Jun 30, 2020 at 9:18
  • $\begingroup$ @Orokusaki For Riemann integral I am using the closed interval. For Lebesgue integral there is no difference between open and closed intervals since every single point has Lebesgue measure $0$. If you want you extend the definition of my function by taking $f(x) =-\ln x$ for $0 <1 \leq 1$ and $f(0)=0$. $\endgroup$ Jun 30, 2020 at 9:24

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