Vertical tangent
= Vertical tangent
A <tangent line> parallel to the vertical coordinate axis. For a regular <level set> $\Phi(x,y)=C$, the conditions $\partial\Phi/\partial y=0$ and $\partial\Phi/\partial x\ne0$ make the <tangent line> vertical: the relation $d\Phi=0$ forces $dx=0$. The <implicit function theorem> then expresses $x$ locally as a differentiable function of $y$, although a graph $y(x)$ has no finite derivative there. The <curve> $x=y^2$ at the origin is a simple example.