Answer
The composition $f\circ g$ is discontinuous only at the point $(0,0)$.
Work Step by Step
The function $g(x,y)=x^2+y^2$ is continuous as a sum of continuous functions $x^2$ and $y^2$.
The function $f(t)=1/t^2$ has a discontinuity at $t=0$.
This means that the composition
$$f\circ g(x,y)=f(g(x,y))$$ is discontinuous when $g(x,y)=x^2+y^2=0$ which is when both $x=0$ and $y=0$. So the composition is discontinuous only at the point $(0,0)$.
