Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]

Question

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...

Answer 1

$f(x)=0,\ x\in{\Bbb R}\backslash{\Bbb Q}$

$f(r/s)=s$, $r/s$ irreducible fraction.

Answer 2

Hint: What if the function gets arbitrarily large on a dense subset of $[a,b]$ of measure zero?

Answer 3

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.

Answer 4

Pick an enumeration of the rationals, and define your function to be $ n $ at the $ n $-th rational and $0$ on all the irrationals. Then this function is zero almost everywhere, but as every non-trivial interval $ I$ contains infinitely many rationals, it must also be unbounded on each such $ I $.