fa.functional analysis – Lower bounds on translates of a function over a compact set

Let $fin L^p(mathbb{R})$ and define $f_theta(x)=f(x-theta)$. Let $Ksubsetmathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following:
$$
inf_{thetanetheta’in K}frac{Vert f_theta – f_{theta’}Vert_p}{|theta-theta’|}.
$$

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.