Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b] Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]. 
*I'm thinking about this but without progress...

Pick a countable, dense subset $\{x_k\}$ of $[a,b]$, and set 
$ f = \sum_{k=1}^\infty \frac{2^{-k}}{\sqrt{|x-x_k|}}$.
Each summand is measurable and positive, so $f$ is a monotone limit of measurable functions. Moreover, if you integrate $f$ you can use the monotone convergence theorem to exchange the sum and integral. Finally, since
$\int_a^b\frac{dx}{\sqrt{|x-x_k|}} \le 2\sqrt{b-a}$,
you find that $\int_a^b f\hspace{2pt}dx$ is finite.
