Gronwall's Lemma (Integral Form)

Let $T$ be a real number with $T>0$, let $[0,T]$ be the closed interval determined by $0$ and $T$, regarded as a subset of the real line $(\mathbb{R},d_{\mathbb{R}})$, and let the codomain $\mathbb{R}$ carry the same metric $d_{\mathbb{R}}$. Write $(0,T]$ for $\{t\in\mathbb{R}:0<t\le T\}$. Let $u:[0,T]\to\mathbb{R}$ be continuous on $[0,T]$, and let $a$ and $b$ be real numbers with $b\ge0$.

For every $t\in(0,T]$ the restriction of $u$ to $[0,t]$ is Riemann integrable on $[0,t]$ by claim 1 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval; write $\int_0^t u(s)\,ds$ for that Riemann integral, and set $\int_0^0 u(s)\,ds=0$.

Suppose that

Then, with $\exp$ the exponential function,