Gronwall's Lemma (Integral Form)

Let $T>0$ be a real number, let $u:[0,T]\to\mathbb{R}$ be continuous on $[0,T]$, and let $a$ and $b$ be real numbers with $b\ge0$. For $t\in(0,T]$ let $\int_0^t u(s)\,ds$ denote the Riemann integral of the restriction of $u$ to $[0,t]$, which exists by Continuous Functions on a Closed Interval are Riemann Integrable, and set $\int_0^0 u(s)\,ds=0$. Suppose that

Then, with the exponential function,