Is the Implicit Function Theorem related to the uniqueness theorem for ordinary differential equations? Is it possible to use the implicit function theorem to prove the existence of ordinary differential equations?
I have seen a proof for the existence of ordinary differential equations in which the conditions are that:
We consider an initial value problem where a function and it's partial with respect to y are continuous. 
However these are two of the conditions for the implicit function to work. So I feel like there should be a way to connect the two.
 A: It is possible to use the implicit function theorem to prove uniqueness of solutions of equations of certain type. The connection is more transparent for systems of the form 
$$ \frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y) \tag{1}$$
which include the standard ODE $y'=g(t,y)$ as a special case (let $f\equiv 1$, so that $x\equiv t$).
Suppose there is a function $\psi$ of $x,y$ such that $\nabla \psi = (-g,f)$. Then (1) implies 
$$\frac{d}{dt}\psi(x(t),y(t)) = 0$$
and therefore every trajectory of (1) is contained in a level set of $\psi$. If the Implicit Function theorem applies to $\psi$, i.e., if it's $C^1$ smooth with nonzero gradient, then the level sets are smooth curves.  In particular, they do not branch, and this implies uniqueness for the solution of (1). 
The existence of $\psi$ (locally) is equivalent to the equality $-g_y=f_x$, which simply says that $(f,g)$ is a field of zero divergence. So, (1) has a unique solution (for given initial values) under this condition. 
One can generalize by observing that it suffices to have $\nabla \psi = (-g,f) h$ with some scalar function $h$.
Closely related: Bendixson–Dulac theorem.
