اگر $f\left( x \right)=\left\{ \begin{matrix}
x+a\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,x\lt 1 \\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,x\ge 1 \\
\end{matrix} \right.$ و $g\left( x \right)=\left\{ \begin{matrix}
x+1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,x\lt 1 \\
\frac{a}{x+1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,x\ge 1 \\
\end{matrix} \right.$، به ازای کدام مقدار $a$ تابع $f+g$ در $x=1$ پیوسته است؟