How do we detect infinitely narrow emission / absorption lines within continuous electromagnetic spectra?

Question

This always bothered me, especially in the case of absorption lines. for instance, if you have a blackbody emitting a continuous spectrum, and then a filter in front that only filters out one very specific frequency or color, how does that actually reduce the total amount of light? it would seem that almost zero (actually an infinitesimally small amount) of light would actually fit that criteria of being exactly the right frequency.

There are 3 explanations I can think of, but none make a whole lot of sense:

1. Emission and absorption lines aren't actually discreet, and will block out a narrow band of frequencies (despite everything I have been taught)
2. Spectra aren't actually continuous, and will therefore be limited to a finite number of colors within a certain range of colors (say, there are only 10 million colors between blue and green and not infinite) but this goes even more against everything I have been taught.
3. Wobbles and ripples in spacetime will make the frequency of a photon passing through this space be not quite constant, and as its frequency oscillates slightly, it is much more likely to, at some point, have the exact frequency that is needed for it to be absorbed.

Are any of these correct? or is it something else, or something about how we measure spectra?

Answer 1

1) is correct. Absorption and emission lines are not infinitely narrow (although they are quite narrow compared to the center frequency).

Actually, the width (or bandwidth) is depending on the lifetime of the state. The longer the lifetime the narrower the line.

This is also clear from a mathematical point of view. The only signal of zero bandwidth is an (infinitely long (!)) sine wave. If the sine wave is only finitely long (e.g. when it is turned on or off), its Fourier transform has a finite width. Since in the physical world everything has a finite duration, zero bandwidth signals only exist in the mathematical "space".

And yes, bandpass filtering blackbody radiation yields a power (approximately) proportional to the bandwidth. Also note that the same argument as above also applies to the filter. A practical filter cannot have zero bandwidth as it would need to have an infinite memory.