$f\left(x\right)$ is continuous on $R$

$\Rightarrow f\left(1^{-}\right)=f\left(1\right)=f\left(1^{+}\right)$

$\left|a+1+b\right|=\underset{x\to 1^{+}}{\lim }\sin \left(\mathrm{\pi x}\right)$

$\left|a+1+b\right|=0\Rightarrow a+b=-1$

$\Rightarrow$ Also

$f\left(-1^{-}\right)=f\left(-1\right)=f\left(-1^{+}\right)$

$\underset{x\to -1}{\lim }2\sin \left(\frac{-\pi x}{2}\right)=\left|a-1+b\right|$

$\left|a-1+b\right|=2$

Either $a-1+b=2$ or $a-1+b=-2$

$a+b=3 ...\left(2\right)$ or $a+b=-1 ...\left(3\right)$

from $\left(1\right)$ and $\left(2\right)$ $\Rightarrow a+b=3=-1$ (reject) 

from $\left(1\right)$ and $\left(3\right)$ $\Rightarrow a+b=-1$