Implicit Function/Examples/x^3 + y^3 - 3 x y = 0/Proof

Proof
This is the Cartesian form of the equation for the folium of Descartes:
 * $x^3 + y^3 - 3 a x y = 0$

for $a = 1$:


 * FoliumOfDescartes.png

It is seen that when $x \le 0$, $y$ is uniquely determined by $x$.

Also, from Maximum Abscissa for Loop of Folium of Descartes, $y$ is also uniquely determined by $x$ when $x > 2^{2/3}$.

In between those values, for $y$ to be defined as a function of $x$ it is necessary to choose one of the $3$ possible values that $y$ can take for each $y$.

For example:


 * FoliumOfDescartes-section1.png $\quad$ FoliumOfDescartes-section2.png $\quad$ FoliumOfDescartes-section3.png

Thus we have:
 * $y = \map f {x, \map g x}$

where $\map g x$ is not straightforward to define.

This demonstrates that $y$ is an implicit function of $x$.