= Solution
{c}
<Continuous dependence of an ODE solution on parameters> here means continuity of the map $\beta\mapsto y_\beta$ into $C([a-h,a+h])$ with its <supremum norm>: for every $\beta_0\in I$ and $\varepsilon>0$, all sufficiently close $\beta\in I$ satisfy
$$
\sup_{|x-a|\leq h}|y_\beta(x)-y_{\beta_0}(x)|<\varepsilon.
$$
Subtract the two <integral equations> and use the uniform <Lipschitz constant> $L$ on the rectangle. With $v(x)=|y_\beta(x)-y_\gamma(x)|$ we obtain
$$
v(x)\leq|\beta-\gamma|\,|x-a|+L\left|\int_a^x v(s)\,ds\right|.
$$
Part (i), with $c=|\beta-\gamma|$ and $A=L$, therefore proves, for $L>0$,
$$
\boxed{\|y_\beta-y_\gamma\|_\infty\leq\frac{e^{Lh}-1}{L}|\beta-\gamma|.}
$$
The case $\beta=\gamma$ follows from uniqueness or the same estimate with zero forcing. If $L=0$, the direct <integral> estimate gives $\|y_\beta-y_\gamma\|_\infty\leq h|\beta-\gamma|$. In either case the map is even <Lipschitz continuous>, and hence <continuous>, throughout $I$ in the stated <supremum norm>.
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