Existence

If $f$ is continuous on an open rectangle

in the $x,y$ plane, and let $(x_0,y_0)\in \mathcal{R}$ be a fixed point in the rectangle.
Then, the IVP

has at least one solution $y(x)$ defined for $x$ in some open subinterval of $(a,b)$ that contains $x_0$.

Uniqueness

In addition, if $\frac{\partial f}{\partial y}(x,y)$ is continuous on $\mathcal{R}$, then the solution is unique.