$f(x, y)=\left\{\begin{array}{ll}x^{2}+y^{2} & \text { if } x^{2}+y^{2}<1 \\ 1 & \text { if } x^{2}+y^{2} \geq 1\end{array}\right.$
Is this function continuous ? It seems like it is circle of radius 1 which is continuous. But I am not sure about it and I have seen this type of continuity problem first time.
$x^2+y^2 \lt 1$ inside the unit circle centered at the origin. You can check that $\lim_{(x^2+y^2)\to 1^-} f(x,y)=1$, and so the function is indeed continuous.
Notice that the curve is a paraboloid inside the circle $x^2+y^2=1$ and the plane $z=1$ otherwise.