Multivariable continuity in ball How do you show continuity for ball-based functions such as
$$f:B[(0,0),1)]\rightarrow\mathbb{R}, \space f(x,y) = \sqrt{(1-(x^2+y^2)}$$

A direct proof: Denote our ball by $B$. We observe that $f(B)=[0,1]$. Let $0 \leq a < b \leq 1$. Define $I = (a,b)$ (open interval) and consider $A = f^{-1}\left( I \right) $. Observe that $$A = \left\{x \in \mathbb{R}^2 \: \middle| \: \sqrt{1- a} <  \left| x \right| < \sqrt{1- b} \right\}$$ which is open. Thus the preimage of an open interval (and consiquently an open set) is open and $f$ is continuous.
You could also say that $f = g \circ h$, where $g(x) = \sqrt{x}$ and $h(x) = 1- (x^2+y^2)$. Thus $f$ is a composition of functions that are continuous  in their domain and thus continuous itself.
