I have been told most real options are American.
However, this isn't really true. Markets are closed at times, there are delays in transactions, or the owner of the option might be sleeping, or just otherwise not keeping up with the markets. So real life behavior is very discrete.
So why are we so much interested in pricing options where you can exercise at any point in $[0, T$]? Especially when that problem is very hard?
Is it not far more realistic to be interested in Bermudan options with e.g. daily-monitoring (252 days per year)?
Bermudan options can also be far more easily priced than Americans. Just use backwards induction with a stepsize $\Delta = 1/252$.
An American price would be obtained if the stepsize $\Delta \rightarrow 0$, but that is very hard to calculate in practice.
So if Bermudan options are more realistic and easier to price, why care about American options that seem mostly a unrealistic mathematical construct?