We propose and experimentally demonstrate a novel switchable and discretely tunable comb filter based on a thermally induced linearly chirped fiber Bragg grating. Experimentally we achieved a thermally induced optical bandpass filter that has eight switchable passband peaks with peak separations of 1.6 and 3.2 nm, a very narrow bandwidth (as small as 10 pm) of each peak, a tunable range of 16.5 nm, and a greater than 25 dB rejection ratio. Two spectral peaks separated by only 0.8 nm were also obtained with this comb filter. This filter provides the unique advantages of an all-fiber structure, switchable spectral peaks, independent tuning of the center wavelength and wavelength spacing of the spectral peaks, low polarization sensitivity, simple structure, ease of fabrication, and low cost.