(1)

\begin{align*} \left\lceil z\right\rceil +\left\lfloor z\right\rfloor & =z-\mod\left(z,-1\right)+z-\mod\left(z,1\right)\\ & =2z-\left(\mod\left(z,1\right)-\mod\left(-z,1\right)\right)\\ & =2z-\left(\mod\left(\Re z,1\right)-\mod\left(-\Re z,1\right)+i\left(\mod\left(\Im z,1\right)-\mod\left(-\Im z,1\right)\right)\right)\\ & =2z-\left(\mod\left(2\Re z,1\right)+\mzp_{0,1}\left(0,1;2\mod\left(\Re z,1\right)\right)-\left|\sgn\mod\left(\Re z,1\right)\right|+i\left(\mod\left(2\Im z,1\right)+\mzp_{0,1}\left(0,1;2\mod\left(\Im z,1\right)\right)-\left|\sgn\mod\left(\Im z,1\right)\right|\right)\right)\\ & =2z-\mod\left(2z,1\right)-\mzp_{0,1}\left(0,1;2\mod\left(z,1\right)\right)+\left|\sgn\mod\left(\Re z,1\right)\right|+i\left|\sgn\mod\left(\Im z,1\right)\right|\\ & =2z-\mod\left(2z,1\right)-\mzp_{0,1}\left(0,1;2\mod\left(z,1\right)\right)+\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \end{align*}

(2)

\begin{align*} \left\lceil z\right\rceil -\left\lfloor z\right\rfloor & =z-\mod\left(z,-1\right)-\left\{ z-\mod\left(z,1\right)\right\} \\ & =-\mod\left(z,-1\right)+\mod\left(z,1\right)\\ & =\mod\left(z,1\right)+\mod\left(-z,1\right)\\ & =\mod\left(\Re z,1\right)+\mod\left(-\Re z,1\right)+i\left\{ \mod\left(\Im z,1\right)+\mod\left(-\Im z,1\right)\right\} \\ & =\left|\sgn\mod\left(\Re z,1\right)\right|+i\left|\sgn\mod\left(\Im z,1\right)\right|\\ & =\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \end{align*}

(2)-2

実数の場合
\begin{align*} \left\lceil x\right\rceil -\left\lfloor x\right\rfloor & =T\left(x\notin\mathbb{Z}\right)\\ & =F\left(x\in\mathbb{Z}\right)\\ & =\left|\sgn\mod\left(x,1\right)\right|\\ & =\sgn\mod\left(x,1\right) \end{align*}