Let $t_0 < t_1$, let $U_c \subset \mathbb{R}^m$ be compact, and let $f: [t_0,t_1] \times \mathbb{R}^n \times U_c \to \mathbb{R}^n$ be continuous. Suppose that there exist constants $a,b \geq 0$ such that, for every $(t,x,u) \in [t_0,t_1] \times \mathbb{R}^n \times U_c$,
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\begin{align*}
|f(t,x,u)| \leq a + b|x|.
\end{align*}
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Let $x_0 \in \mathbb{R}^n$. For each $k \in \mathbb{N}$, let $u_k: [t_0,t_1] \to U_c$ be Lebesgue measurable, and let $x_k \in AC([t_0,t_1];\mathbb{R}^n)$ satisfy
for $\mathcal{L}^1$-almost every $t \in [t_0,t_1]$. Then the family $(x_k)_{k \in \mathbb{N}}$ is uniformly bounded on $[t_0,t_1]$ and equicontinuous on $[t_0,t_1]$. More precisely, with $T := t_1 - t_0$ and
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\begin{align*}
R := (|x_0| + aT)e^{bT},
\end{align*}
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one has $|x_k(t)| \leq R$ for every $k \in \mathbb{N}$ and every $t \in [t_0,t_1]$, and