Lp spaces - example of function

Question

Can you give me an example of function which: $$f \in L^{p}[a,b]$$ but $$f \not\in L^{\infty}[a,b]$$ $L^{\infty}[a,b]$ is space of essentially bounded function at interval $[a,b]$

$1 \le p < \infty$

Thanks a lot.

Answer 1

The function $x\to-\log x$ belongs to $L^p((0,1))$ for any $p\geq 1$ but it is not essentially bounded.