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Continuous absorption is the absorption of more than mere individual spectral lines: absorption over a range of wavelengths. Any opacity over some wavelength-range falls under this definition, e.g., the gaps between Earth's atmospheric windows. It can occur if line broadening is such that gaps between a bunch of adjacent absorption lines don't exist, or it can be simulated if the resolution of an instrument is such that the individual lines of a substance cannot be distinguished. In either case what is viewed is some reduction in the electromagnetic radiation over a range of wavelengths (line blanketing), both cases common for molecular absorption: molecules often have many closely-spaced energy levels and resultant absorption lines. Other processes that can produce it include photoionization, which can reduce the intensity at all wavelengths corresponding to photons of sufficient energy to ionize the material (e.g., creating the Balmer jump).
Band absorption refers to such absorption over a limited range of wavelengths, which can be referred to as an absorption band.