Given the function

$f[f(x)]=\frac{1}{\frac{1}{x+2}+2}=\frac{x+2}{2 x+5}$

So, $f[f(x)]$ is not defined at $x+2=0$ and $2 x+5=0$

If $x+2=$, then $x=-2$

If $2 x+5=0$, then $x=-\frac{5}{2}$

Hence, the function is discontinuous at $x=-\frac{5}{2}$ and $-2$