When I was studying dispersion of refraction index in semiconductors and dielectrics, my professor tried to explain that if a filter (like a dielectric absorbing some light frequencies, or an electric RC-filter) removes some frequencies, then the remaining ones must be phase shifted to compensate for those frequencies (which are infinitely spread in time as usual monochromatic signals) being subtracted from the whole signal, to preserve causality.
I intuitively understand what he was talking about, but what I'm not sure of is whether his argument is really justified - i.e. whether there can exist a non-trivial filter, which absorbs some frequencies and leaves remaining ones not shifted, but still preserving causality. I can't seem to construct one, but can't prove it doesn't exist as well.
So the question is: how can it be (dis)proved that a causal filter must shift phases of frequencies relative to each other?