Showing that $f$ is $C^\infty$ Question:

Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2  +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$
  Show that $f$ is of class $C^\infty$.

Attempt:
Define $F(x,y,z) = (x^2 + y^4)z + z^3 - 1$. Then $$F_z(x,y,z) = (x^2 + y^4) + 3z^2 > 0$$
for any $(x,y,z) \in \mathbb R^3- \{0\}$, and if we fix $(x_0,y_0) \in U$, such that $(x_0,y_0) \neq 0$, then taking $z_0 = f(x_0,y_0) \in \mathbb R$, thus it follows that $F(x_0,y_0,z_0) = 0$. 
Now by the Implicit Function Theorem we have that $z$ is a defined as a function of $x$ and $y$, such that $$F(x,y,z) = F(x,y,\xi (x,y)) = 0 \tag{1}$$ for every $(x,y,z) \in B \times J$, here $B \subset \mathbb R^2$ and $J \subset \mathbb R$. Clearly $F$ is of class $C^\infty$ then $z = \xi (x,y)$ is of class $C^\infty$. 
As $f$ is continuous there exists $\delta > 0$ sufficiently small such that $f(B) \subseteq J$, then we may conclude by $(1)$ and by hypothesis $F(x,y,f(x,y)) = 0$, that  $f(x,y) = \xi(x,y)$, for every $x \in B$, it follows then that $f$ is of class $C^\infty$. 

The discriminant of $P(z) = t z + z^3 - 1$ is $-4 t^3 - 27$, which is nonzero for $t \ge 0$.  Thus $\dfrac{\partial}{\partial z} P(z) \ne 0$ whenever $P(z) = 0$ with $t \ge 0$.  In particular, this applies when $t = x^2 + y^4$.
