A continuous real valued function on $\mathbb{R}$ with compact support is said to belong to the Zygmund class, $f \in \Lambda_\ast,$ if \begin{equation} \sup_{x,h\in\mathbb{R}} \frac{|f(x+h)+f(x-h)-2f(x)|}{|h|} < \infty. \end{equation} It is known that the space $\mathrm{I}(\mathrm{BMO})$ of functions with $\mathrm{BMO}$ derivative in the distributional sense is a subspace of $\Lambda_\ast.$ In this talk, based on a joint work with A. Nicolau, we give an estimate for the distance of a given function $f \in \Lambda_\ast$ to the subspace $\mathrm{I}(\mathrm{BMO}).$ We will do so by means of a discretisation similar to another used previously by J. Garnett and P. Jones to study the space $\mathrm{BMO}.$