In Glendinning's book Stability, Instability and Chaos (theorem 1.2), he said that if in the ode $\dot x = f(x,t)$, $f$ is smooth near $(0,0)$, and the initial condition $x(0)\in(-\epsilon, \epsilon)$, then solutions $x(t)$ exist and depend continuously on the initial condition $x(0)$.
This statement seems to negate the possibility of chaos (i.e. solutions being sensitive to $x(0)$) for such an ode. But the author says it is not so, because 'solutions are' not necessarily 'uniformly continuous in both variables...space and time' $x(0), t$, 'double limits can be tricky.'
The author seems to say the reason is that $f$ is not necessarily continuous in (x(0),t). For example, for $\dot x = x$, $x(C,t) = C e^t$, even if this function is continuous on $C$ and on $t$, it may not be a continuous function of $(C,t)$ , respectively. (This is perhaps not a proper example. But I hope the readers see my point here.) But the author's explanation is fairly sketchy. How to explain the possibility of chaos for such a function, in detail?