The dielectric constant $\epsilon$ is a quantity (a number) which appears in electroSTATIC when people describe how a material screens an external TIME-INDEPENDENT electric field. When they begin to study how a material screens an external TIME-DEPENDENT electric field $E\propto\exp(-i\omega t) (electroDYNAMICS) they found that the number $\epsilon$ depends on the frequency $\omega$, so one gets $\epsilon(\omega)$. It would be stupid to call a quantity, which essentially depends on frequency, just "dielectric constant" (although the physical meaning is the same), therefore one calls it "dielectric function". Further studies showed that $\epsilon$ depends not only on the frequency but also on the wave-vector of the field, $E\propto\exp(iqx-i\omega t), so one gets the dielectric function $\epsilon(q,\omega)$. If q=0 and omega=0, the dielectric function becomes the dielectric constant; one can use, again, this term.

Dielectric permittivity = dielectric function (or constant).

Yes, the dielectric function is the same as the function of the (relative) permittivity.

The dielectric constant $\epsilon$ is a quantity (a number) which appears in electroSTATIC when people describe how a material screens an external TIME-INDEPENDENT electric field. When they begin to study how a material screens an external TIME-DEPENDENT electric field $E\propto\exp(-i\omega t) (electroDYNAMICS) they found that the number $\epsilon$ depends on the frequency $\omega$, so one gets $\epsilon(\omega)$. It would be stupid to call a quantity, which essentially depends on frequency, just "dielectric constant" (although the physical meaning is the same), therefore one calls it "dielectric function". Further studies showed that $\epsilon$ depends not only on the frequency but also on the wave-vector of the field, $E\propto\exp(iqx-i\omega t), so one gets the dielectric function $\epsilon(q,\omega)$. If q=0 and omega=0, the dielectric function becomes the dielectric constant; one can use, again, this term.

Dielectric permittivity = dielectric function (or constant).

Usually when the permittivity of a material is function of space or frequency, it is called dielectric function. This is the case for plasmonic materials (gold, silver at optical frequencies) where their permittivity is function of frequency and usually called dielectric function.

Dielectric function-dielectric constant as a function of frequency and a wave number vector.

Is there any relation between the dielectric components of a material (real and imaginary part) with its absorption coefficients ?

Hussein Ali Jan Miran Yes, there is! For isotropic media, the square of the complex index of refraction function is equal to the dielectric function. The imaginary part of the complex index of refraction function is the absorption index function k. If you multiply 4*Pi*k/(ln10) with the wavenumber then you obtain the absorption coefficient.

See e.g. Article Beer's Law – Why Absorbance Depends (Almost) Linearly on Concentration

Many thanks go to you dear Thomas. Your explanation is highly appreciated.

Cheers