Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has sides parallel to axis) is defined as the following integral: $\frac{1}{|Q|}\int_{Q}|u(y)-u_Q|\,\mathrm{d}y$,
where $|Q|$ is the volume of $Q$, i.e. its Lebesgue measure, $u_Q$ is the average value of $u$ on the cube $Q$, i.e. $u_Q=\frac{1}{|Q|}\int_{Q} u(y)\,\mathrm{d}y$.
A BMO function is any function u belonging to $L^1_{\textrm{loc}}(\mathbb{R}^n)$ whose mean oscillation has a finite supremume over the set of all cubes $Q$ contained in $\mathbb R^n$.
I could find many examples for functions in BMO. But I could not find a function $u:\mathbb R^n\to\mathbb R$ which is not constant, so that $u$ in BMO and $u(tx)=u(x)$ for almost everywhere $t\in[0;1]$, and for every $x\in \mathbb R^n$.
I also want to find such function $u$ in BMO so that $u(tx)=u(t)$ for almost every $t\neq0$, and every $x$
So my question is that: does exits such function $u$, and could you give me any example.