Indeed you can't, at least for ODE in $\mathbb{R^n}$: Peano's theorem asserts that any Cauchy problem for the ODE $\dot u(t)=f(t,u)$ with continuous non-linearity $f$, admits local solutions on some interval $I=[a,b]$, , which are a connected compact set in $C^0(I,\mathbb{R^n})$. Note that in that generality ($f$ continuous) it is sufficient to consider autonomous equations. One can see this theorem as an instance of Schauder's fixed point theorem, or also prove it via approximation by Lipschitz problems with $f_k\to f$, using the Ascoli-Arzelà theorem to obtain a convergent subsequence of the solutions of the approximated problems.
The analogous equation $\dot u(t)=f(u)$ with continuous $f$ in a Banach space, e.g. $\ell_2$, may have no solution at all. The idea for a counterexample is coupling countably many scalar equations $\dot u_n(t)=f_n(u)$ with blow-ups at $T_n\to 0$ in such a way that the resulting$f$ be continuous. Check e.g. Dieudonné's Foundations of modern Analysis for such a counterexample.