For Problem Z-1: $\mathbb{Z}$ satisfies $(\exists x)(\exists y > x)(\forall z)(x \geq y$    or $y \geq z)$.

For Problem Z-2: $(\exists x)(\exists \epsilon > 0)(\forall y)(\vert x-y\vert< \delta$    and $\vert f(x) - f(y)\vert \geq \epsilon)$.