Example of function in $L^1(S^1,\mathbb{R})$ but not in $\operatorname{BMO}(S^1,\mathbb{R})$

Question

I am searching an example of a function $f:S^1\rightarrow \mathbb{R} \in L^1$, but $f \notin \operatorname{BMO}$.

Where BMO means Bounded Mean Oscillation https://en.wikipedia.org/wiki/Bounded_mean_oscillation

How do I construct such a function?

Yours, Maxi

Answer 1

Hint: Look at $x^{-1/2}$ on $(0,1).$ In particular, consider

$$\frac{1}{h}\int_0^h \left |\,\,x^{-1/2} - \frac{1}{h}\int_0^h t^{-1/2}\,dt\,\,\right |\, dx.$$