# 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.