Range of a function of two variables given by two sentences I was given the following function:
$$f(x, y) := \begin{cases}
             2 - \frac{1}{x^2\,+\,y^2}  & \text{if } 0<x^2 + y^2 \leqslant 1 \\
             \sqrt{x^2 + y^2-1}  & \text{if } 1 < x^2 + y^2
       \end{cases} \quad$$
How can I find it's range? 
I'm conviced it is the interval $(0, 1]$, by simple calculation. 
Is it right? It's simple, but I'm a little concerned about the disjoint intervals of the ranges of the piecewise functions.
Thanks in advance! 

Hint:
I'm not sure why you consider the range is $(0,1]$. Instead, for the first part, consider what happens to $f(x,y) = 2 - \frac{1}{x^2 + y^2}$ as $x^2 + y^2 \to 0^{+}$. For the second part, consider what happens to $f(x,y) = \sqrt{x^2 + y^2 - 1}$ as $x^2 + y^2 \to \infty$.
