Application of implicit function theorem?

The implicit function theorem would apply directly to tell you that if you have the equation $$F(x,y,z) = (x^2+y^4)z + z^3 = 1,$$ then you can locally solve for $z=f(x,y)$ near $(x_0,y_0,z_0)$ as a $C^1$ function provided

(1) $F$ is $C^1$ (it is)

(2) $\partial F/\partial z \ne 0$ at the point $(x_0,y_0,z_0)$.

What do you notice about $\partial F/\partial z$ at every point in $U$?

As usual, Ted gave a great answer (and you should accept it). I'll just nitpick a bit, which you might or might not appreciate now: being $C^1$ is a local property, so you want to check that given $(x_0,y_0)\in U$, then $f$ is of class $C^1$ in a neighbourhood of $(x_0,y_0)$.

Great, so once you apply the IFT to the funcion $F$ in Ted's answer, you get a neighborhood $V$ of $(x_0,y_0)$ (which you can assume it's contained in $U$, replacing $V$ by $V\cap U$ if necessary) and an open interval $I$ centered in $z_0$ such that for all $(x,y) \in V$ there is a unique $\varphi(x,y)\in I$ such that $$F(x,y,\varphi(x,y))=1, $$and this implicit function $\varphi:V\to I$ is also $C^1$. By the uniqueness above, $f|_V= \varphi$ is $C^1$. And that's the reason $f$ is $C^1$. I just wanted to ilustrate the importance of the uniqueness part of the theorem, so you can use its full power without missing anything.