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The accidental depends on the resolution. These usually correspond to the chromatic scale. If it resolves upward then its raised.
Of course it isn't always done that way since it can be hard to read(such as when double accidentals come into play) or in special cases when form comes into play(notable one is Beethoven's 7th symphony where its written as C# but he writes it as a Db intentionally because of what happens on the recapitulation IIRC).
For example, if you have a passing tone like C C#/Db D then you don't write C Db D since its obvious that the Db is moving upward and hence it is a C#.
Now its true that for fixed pitch instruments Db and C# are the same it is not true for non-fixed pitched instruments. In this case Db and C# are truly different and C C# D will sound different than C Db D(although most people cannot hear the difference). So in reality these are two different notes and hence have different functions. Hence we should retain this in fixed pitch instruments because they might be played on non-fixed pitch instruments.
Although there are theoretical reasons to notate them properly too. For example, If you have the progression Cmaj G7 Cmaj you obviously don't write the G7 as F##7 because its functioning as a G and not an F##7(assuming we are in the key of C).
If, say, you have a modulation from C to G then its the same thing... but in some cases, as has been pointed out, you might make a enharmonic modulation because it is easier to read. e.g., maybe modulating from E to C#m could be reinterpreted as E to Dbm and notated that way because, say, we might make end up wanting to modulate to C# later.
E.g., E-> Db#m -> Db or E -> C#m -> Db instead of E -> C#m -> C# both which hide whats really going on.
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