Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator. How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$ (in $L^{+\infty}$ norm) then $f=g$ almost everywhere, where $g$ is continuous?

I ask for hints or ideas only :-)

I tried use something like modulus of continuity:

$\omega(f,\delta)=\mathrm{esssup}_{|x-y|<\delta} |f(x)-f(y)|$ and show that function $f$ must be uniformly continuous in the set of full measure, next extend from the dense set to entire $\mathbb{R}$. I didn't see, however, how to use separability...