  Let $(X,f)$ be a dynamical system with the specification property and
$\varphi$ be a continuous function. In this paper, we consider the multifractal
irregular set
  \begin{align*}
  I_{\varphi}=\left\{x\in
X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^ix)\ \text{does
not exist}\right\}
  \end{align*} and show that this set is either empty or carries full Bowen
upper and lower metric mean dimension with potential.
