Can someoe help with this : What is the precise arbitrage argument demonstrating that the price of an american option should be continuous around an ex-dividend date?
If in the domain considered it is never optimal to exercise, then the price of an American option converges to the price of an European option. This means that the price of an American option is continuous in a domain of non-exercise as it is the case for European Options (1). This also means that the price of an American Option could be replicated by a portfolio of European options, one for each of the time-segment where early exercise is never optimal.
(1) I'd like to point out that this is, afaik, true for any model as option models give continuous prices for European products by design (with proper parameter choice) - it would be a trading/hedging hell otherwise.
The only time the holder of an American call option should consider early exercise is just prior to the stock going ex-dividend. There is where you may have your discontinuity as at exercise opportunity the value of American call is the max between the value of the product shall the owner not exercise and the value of the exit opportunity. Mark Joshi in "The concepts and practice of Methematical finance" explains better this, here I am just trying to pass along an intuition. The idea is to work backwards, from maturity to T=0, looking at the European replication and at the exercise opportunity work out the max. Similar argument applies to American puts where we could create a mesh of intervals where it is never optimal to exercise between points of possible early exercise, that over the entire life-time of the option.