Does a filter need to completely attenuate high frequencies to be considered low-pass? I'm looking at a system with a frequency response that is: $$H(e^{j\omega})=\tfrac{1}{4}(2\cos(\omega)+\cos(2\omega)+2)$$

I think the magnitude of this system looks like this (not sure what is the proper name for a function like this):



Would this be considered a low-pass filter even though the output is always greater than zero? If not, what would be the proper name for it?
Ideal low pass filter is supposed to attenuate/remove all the frequencies above the cutoff. but in practice the filters have non zero magnitude even for frequencies in their attenuation region, still they are called low pass.

if the magnitude of filter is quite high for frequencies above cutoff it might be called bad lowpass filter, but is still a low pass because it attenuates the frequencies above cutoff.