Neither Open nor Closed Sets via Preimage Argument

Question

Show that the set below is neither open nor closed. Do so by finding $f^{-1}(S)$ for the function $f$. Find a suitable function $f$ if no function is given.

$$S:=\left\{(x,y,z)\in\Bbb R^3~\middle|~ \sqrt{(x^2+y^2)}\leqslant z<1\right\}$$

Answer 1

$f(x,y,z) = \sqrt{x^2 + y^2} - z$ restricted to $\mathbb R\times \mathbb R\times [0,1)$.