Picard--Lindelöf theorem

From Wikipedia.
Theorem (Existence and Uniqueness Theorem):
Let $D \subseteq R \times R^{n}$ be a closed rectangle with $(t_{0}, y_{0}) \in int D$ , the interior of $D$ . Let $f : D \to R^{n}$ be a function that is continuous in $t$ and Lipschitz continuous in $y$ (with Lipschitz constant independent from $t$ ). Then there exists some $ε > 0$ such that the initial value problem

y^{'} (t) = f (t, y (t)), y (t_{0}) = y_{0}

has a unique solution $y (t)$ on the interval $[t_{0} - ε, t_{0} + ε]$ .

Remarks

If we assume only continuity, the Peano existence theorem assure only existence, not uniqueness.
The theorem can be weakened to
Theorem (Existence and Uniqueness Theorem):
Let $R = [a, b] \times [c, d] \subset R^{2}$ be a rectangle such that the point $(x_{0}, y_{0})$ is in the interior of $(a, b) \times (c, d)$ .
If:

$f (x, y)$ is continuous on $R$ .
The partial derivative $\frac{\partial f}{\partial y} (x, y)$ exists and is continuous on $R$ .
Then, there exists an open interval $I \subset (a, b)$ such that $x_{0} \in I$ , and a unique function $y (x)$ defined on $I$ that satisfies the initial value problem:

y^{'} (x) = f (x, y (x)), y (x_{0}) = y_{0} .

Continuity of $\frac{\partial f}{\partial y}$ is a sufficient condition for Lipschitz continuity.