Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Theorem
Let $D \subseteq \R^2$ be a region containing $\tuple {a, b}$.

Let $f: D \to \R$ be real-valued mapping such that $f$ and $\dfrac {\partial f} {\partial x}$ are continuous on $D$.

Consider the initial value problem:
 * $\dfrac {\d x} {\d t} = \map f {t, x}$
 * $\map x a = b$

Suppose the initial value problem above has a solution $\phi$ for all $x$ in some interval $J$ containing $a$.

Then the solution $\phi$ is unique on $J$.