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When you get some of these results written up, please let me know. I'm quite interested to see them as they are not too far removed from some stuff I did a while ago.
You also need to specify the behaviour of the solution to the differential inequality as $|x|\to\infty$ (namely, how the solution can blow up ... or not blow up). See the link below for a good reason why this is required. Additionally, with regard to max principles you are intending to use, you can probably include spatial inhomogeneity in the coefficients without messing things up too much. mathoverflow.net/questions/82408/…
In your question, it is not clear to me whether or not continuity of $f$ on $[0,1]$ is a requirement ... is it? Otherwise $f:[0,1]\to\mathbb{R}$ given by, $$ f(x) = \begin{cases} 0 &;x\in\mathbb{Q}\cap [0,1] \newline 1 &;x\in[0,1]\backslash \mathbb{Q} \end{cases}$$ may be relevant.
