# Prove integrability of a function

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!!

• By integrable, do you mean Riemann integrable or Lebesgue integrable? Jun 30, 2020 at 9:01
• Riemann integrable Jun 30, 2020 at 9:50

If you are talking about Lebesgue integration this is false. $$-ln x$$ is integrable on $$(0,1)$$ but $$e^{-\ln x}$$ is not integrable.
• @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 $f(0)=0$. Jun 30, 2020 at 9:24