Let $R>0$ be a fixed constant and let $b:\mathbb{R}\to\mathbb{R}$ be locally Lipschitz continuous with polynomial growth, i.e. there are a $D>0$ and $n\in\mathbb{N}$ such that $|b(x)|\leq D(1+|x|^n)$ for all $x\in\mathbb{R}$.
The aim is to find $0<\delta<1$, $C>0$, and $K>0$ such that for all \begin{align} d&\in\mathcal{D}_{C,\delta}=\{d\in\mathcal{C}^\infty([0,1]):\,|d(t)|\leq C\text{ only on intervals of length at most }2\delta\}, \end{align} there are $$ f\in\mathcal{F}_K=\{f\in\mathcal{C}^1([0,1]):\,f(0)=0,\,\|f^\prime\|_\infty\leq K\} $$ and $0\leq t_1\leq t_2\leq 1$ with $t_2-t_1\geq\delta$, for which the solution of the ordinary differential equation $$ x(t)=d(t)+f(t)+\int_{0}^1b(x(s))\,ds,\quad t\in[0,1], $$ satisfies $|x(t)|>R$ for all $t\in[t_1,t_2]$. Notice that the ODE has a unique global solution.
My question is now: Is this possible under the rather general assumptions on $b$?