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?

Answer 1

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.