(1)

\begin{align*} H_{a}\left(x\right)+H_{1-a}\left(-x\right) & =H_{a}\left(x\right)-H_{1-a}\left(x\right)+1+\left(2\left(1-a\right)-1\right)\delta_{0,x}\\ & =\left(a-\left(1-a\right)\right)\delta_{0,x}+1+\left(1-2a\right)\delta_{0,x}\\ & =-\left(1-2a\right)\delta_{0,x}+1+\left(1-2a\right)\delta_{0,x}\\ & =1 \end{align*}

(2)

\begin{align*} H_{a}\left(x\right)+H_{b}\left(-x\right) & =H_{a}\left(x\right)+H_{1-a}\left(-x\right)+\left(b-\left(1-a\right)\right)\delta_{0,-x}\\ & =1+\left(a+b-1\right)\delta_{0,x} \end{align*}

(3)

\begin{align*} H\left(\pm1\right)+H\left(\mp1\right) & =\frac{1\pm1}{2}+\frac{1\mp1}{2}\\ & =1 \end{align*}

(4)

\begin{align*} H_{a}\left(x\right)-H_{a}\left(-x\right) & =H_{a}\left(x\right)-\left(-H_{a}\left(x\right)+1+\left(2a-1\right)\delta_{0,x}\right)\\ & =2H_{a}\left(x\right)-1+\left(1-2a\right)\delta_{0,x}\\ & =2\left(H_{a}\left(x\right)+\left(\frac{1}{2}-a\right)\delta_{0,x}\right)-1\\ & =2H_{\frac{1}{2}}\left(x\right)-1\\ & =2\frac{\sgn\left(x\right)+1}{2}-1\\ & =\sgn\left(x\right) \end{align*}

(5)

\begin{align*} H_{a}\left(x\right)-H_{b}\left(-x\right) & =H_{a}\left(x\right)-\left(H_{a}\left(-x\right)+\left(b-a\right)\delta_{0,-x}\right)\\ & =\sgn\left(x\right)+\left(a-b\right)\delta_{0,x} \end{align*}

(6)

\begin{align*} H\left(\pm1\right)-H\left(\mp1\right) & =\frac{1\pm1}{2}-\frac{1\mp1}{2}\\ & =\pm1 \end{align*}