Continuous, absolutely integrable, unbounded function

Question

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?

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

score 0 · Answer 1 · answered Aug 19 '16 at 15:32

0

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.

answered Aug 19 '16 at 15:32

Zou lang

1

Continuous, absolutely integrable, unbounded function

1 Answers1