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$. 1 You are talking about$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$B^{1,\infty}_\infty\supset L^\infty\cap \text{Lipschitz}$. Now the derivative of$u$belongs to$B^{0,\infty}_\infty$. In particular, derivatives of functions in$L^\infty\cap \text{Lipschitz}$belong to$B^{0,\infty}_\infty\$.