  We consider a recursive system which was introduced by Derrida and Retaux (J.
Stat. Phys. ${\bf 156}$ (2014) 268-290) as a toy model to study the depinning
transition in presence of disorder. Derrida and Retaux predicted the free
energy $F_\infty(p)$ of the system exhibit quite an unusual physical phenomenon
which is an infinite order phase transition. Hu and Shi (J. Stat. Phys. ${\bf
172}$ (2018) 718-741) studied a special situation and obtained other behavior
of the free energy, while insisted on $p=p_c$ being an essential singularity.
Recently, Chen, Dagard, Derrida, Hu, Lifshits and Shi (Ann. Probab. ${\bf 49}$
(2021) 637-670) confirmed the Derrida-Retaux conjecture under suitable
integrability condition. However, in the mathematical review, it is still
unknown whether the free energy is infinitely differentiable at the critical
point. So that, we continue to study the infinite differentiability of the free
energy in this paper.
