You are talking about $B^{0,\infty}_\infty$.
Take a function $u$ in the Zygmund class $B^{1,\infty}_\infty$, which the vector space of $L^\infty$ functions such that $$\exists C,\forall x,h,\quad \vert(u(x+h)+u(x-h)-2u(x)\vert\le C\vert h\vert. $$
Note that $B^{1,\infty}_\infty\supset L^\infty\cap \text{Lipschitz}$. text{Lipschitz}$ (here Lipschitz means $\exists C,\forall x,h,\quad \vert(u(x+h)-u(x)\vert\le C\vert h\vert $).
Now the derivative of $u$ belongs to $B^{0,\infty}_\infty$.
In particular, first derivatives of functions in $L^\infty\cap \text{Lipschitz}$ belong to $B^{0,\infty}_\infty$.
