Question: I am reading "Analysis on Manifolds" by James R. Munkres.

On p.77,

> 
> Example 4. Let $f:\mathbb{R}^2\to\mathbb{R}$ be given by the equation $$f(x,y)=y^2-x^4.$$
> Then $(0,0)$ is a solution of the equation $f(x,y)=0$. Because $\frac{\partial f}{\partial y}$ vanishes at $(0,0)$, we do not expect to be able to solve for $y$ in terms of $x$ near $(0,0)$. In fact, however, we can do so, and we can do so in such a way that the resulting function is differentiable. However, the solution is not unique.
> 
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[![y^2 - x^4 == 0](https://i.stack.imgur.com/WsiRy.png)](https://i.stack.imgur.com/WsiRy.png)

I had the following question:

> 
> Is there an example such that $f(x,y)=0$ at some point and $\frac{\partial f}{\partial y}$ vanishes at the point but we can solve for $y$ in terms of $x$ near the point and the resulting function is differentiable and unique near the point?
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Answer: 
What about
$$f(x,y)=y^3-x^9$$
around the origin? Assuming you are only interested in real solutions.

Or
$$f(x,y)=(x-y)^2?$$
A double root makes the partial derivative vanish :-)
$f(x,y)=y^3-x$ around the origin would work.