Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space $F_{1,\mathrm{\infty }}^{d+1}({\mathbb{R}}^d)$

An evidence of temporal dis-continuity of the solution in $F_{1,\mathrm{\infty }}^s({\mathbb{R}}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed.