Answer
Yes, the function is continuous.
Work Step by Step
Since ${x^2} + {y^2}$ is continuous, so $f\left( {x,y} \right)$ is continuous in the domain ${x^2} + {y^2} < 1$.
Since $f\left( {x,y} \right) = 1$ is constant, so $f\left( {x,y} \right)$ is continuous in the domain ${x^2} + {y^2} \ge 1$.
Therefore, we conclude that $f\left( {x,y} \right)$ is continuous everywhere on ${\mathbb{R}^2}$.
