Let $f:\mathbb{R}\to\mathbb{R}$ be a function (whose properties I shall discuss later) and define $f_\theta(x)=f(x-\theta)$. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'}\frac{\Vert f_\theta - f_{\theta'}\Vert_p}{|\theta-\theta'|}.
$$
In particular, I want to understand how this depends on $f$, and would like a bound that depends explicitly on $f$. This is also where the properties of $f$ come in: The weaker the assumptions the better, but e.g. if there a nice bound that depends (say) on the deriviatives of $f$, then we can assume the needed regularity.

My suspicion is that there is an easy counterexample to show this can be rather poorly behaved even for smooth functions, but I have not been creative enough so far.