= Solution
The \b[<Picard-Lindelöf theorem>] gives local existence and uniqueness for an <initial value problem> when $f$ is continuous on a rectangle around $(a,b)$ and is <Lipschitz continuous> in its second variable there, with a constant uniform in the first variable. A sufficient hypothesis is that $f$ and $\partial f/\partial y$ are continuous on that rectangle.
More quantitatively, if the rectangle is $|x-a|\leq h_0$, $|y-b|\leq k$, and $|f|\leq M$, a sufficiently small $h>0$ with $h\leq h_0$ and $hM<k$ yields a unique differentiable solution on $[a-h,a+h]$ remaining inside the vertical strip. The proof uses the integral equation
$$
y(x)=b+\int_a^x f(t,y(t))\,dt
$$
and <Picard iteration>, or the <Banach fixed-point theorem> after shrinking $h$ to make the integral operator a contraction. The continuity and Lipschitz hypotheses must hold near the actual initial point.
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