Example of bounded function in $L^1[a,b]$ but not in $L^2[a,b]$

Question

Can you give me an example of a bounded function $f\in L^1[a,b]$ which $f\not\in L^2[a,b]$?

Thank you

Answer 1

It is not possible since $$\int_{[a,b]}{|f(x)|^{2} \, d\lambda(x) } \leq M_{f}\int_{[a,b]}{|f(x)| \, d\lambda(x)} < \infty$$

Where $M_{f}>0$ is a bound of $f$

Moreover $$L^{\infty}[a,b] \subset L^{2}[a,b]$$