If $f:U\to \mathbb{R}$ is continuous and $(x^2+y^4)f(x,y) + (f(x,y))^3=1$, then $f$ is $C^\infty$

Question

Let $f:U\to \mathbb{R}$ be continuous in $U \subset\mathbb{R}^2$, such that $$(x^2+y^4)f(x,y) + (f(x,y))^3=1$$ for all $(x,y) \in U$. Prove that $f\in C^{\infty}$.

I'm learning the implicit function theorem, I read it, and its proof, and I thought I had understood it. But when I face a problem like this I just don't know how to use it yet. If someone could help me I would appreciate it.

Answer 1

Hint: Consider $F\colon \mathbb R^3\to\mathbb R$, $F(x,y,z)=(x^2+y^4)z+z^3$. On the level set $F(x,y,z)=1$, when can you solve for $z$ locally as a (smooth) function of $(x,y)$?