We look at the set $C : =\lbrace (x,y) \in \mathbb{R}^2 \mid y^2 = x^2+x^3 \rbrace$.
We want to know if $C$ is a $C^\infty$-manifold. When looking at the plot it is easy to see that $(0,0)$ is the problem here but we are interested in solving this analytically.
With $\lbrace (x,y) \in \mathbb{R}^2 \mid y^3 = x^2 \rbrace$ we can see that a projection $\phi$ onto the $x$-axis is homeomorph but $\phi^{-1}$ is not continuously differentiable. So there is no chart.
How would you argument on $C$ while dont have any idea how the set looks?
It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.
When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=\pm x$, a cross, which is not locally homeomorphic to an interval of $\mathbb{R}$.
For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).