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For my course in Fourier analysis I need an example of a continuous unbounded function that is absolutely integrable. I can't seem to think of one and so need some help with this.

I also need an continuous bounded function that isn't absolutely integrable and I was thinking $f(x)=\begin{cases} \frac{\sin(x)}{x}, \text{if } x\neq 0 \\ 1, \text{ if } x=0\end{cases}$. Is this correct?


marked as duplicate by Alex Provost, Clement C., Behrouz Maleki, Chill2Macht, Joey Zou Aug 19 '16 at 22:34

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  • $\begingroup$ Hint: think of hat / triangle functions whose area are summable but the tips increase to infinity. $\endgroup$ – user251257 Aug 19 '16 at 15:16
  • $\begingroup$ What does "absolutely integrable" mean? By definition, a function is integrable, if the integral if its absolute value exits. $\endgroup$ – M. Rahmat Aug 19 '16 at 15:47
  • $\begingroup$ @M.Rahmat that's for Lebesgue integral. For improper Riemann integral, we do distinguish between conditional and absolute integrability. $\endgroup$ – user251257 Aug 19 '16 at 15:51
  • $\begingroup$ Do you mean absolutely integrable on $\mathbb{R}$ or on a finite interval? $\endgroup$ – DisintegratingByParts Aug 19 '16 at 19:06

Yes,you are right.

First, $\int_{0}^{+\infty} \sin(x)/xdx=\frac{\pi}{2}$,but $\int_{0}^{+\infty} |\sin(x)/x|dx=\infty$.

Second, $\lim_{x\rightarrow 0}\frac{\sin(x)}{x}dx=1$.

So, Your example is correct.


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