Let $C$ be the cone $\{(x,y,z)\in \mathbb{R}:x^2+y^2=z^2,z\geq 0$. Let $g:\mathbb{R}^2\rightarrow \mathbb{R}$ be any differentiable map. Show that the map $\phi:C\rightarrow \mathbb{R}^3$, $\phi(x,y,\sqrt{x^2+y^2})=(x,y,g(x,y))$ is an embedding.
First of all we must prove that $\phi$ is differentiable. I wanted to say that partial derivatives are continuous to show it. But, how could I define them since I have $\sqrt{x^2+y^2}$ instead of $z$?