We are told that the absorption of EM radiation by an atom, resulting

in an electron jumping from one state to another, is very precise and

exact - in terms of the energy absorbed, and the frequency "deleted".

My question is HOW precise? If the phrase "exact energy match" is taken

literally then there will never be ANY noticeable energy taken out (for the

same reasoning as in mathematics, which implies that a precisely define

line has no width at all and therefore contributes ZERO area in an integration).

In other words "lines" in the absorption spectrum will be too narrow to see.

But of course actual absorption spectra do show up.

How come?