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

An evidence of temporal discontinuity of the solution in \(F^s_{1, \infty }(\mathbb {R}^d)\) 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.