# Deriving American option greeks

I am using integral representation of option value instead of trees, so I imagine to derive greeks we have to integrate across time for the boundary to get the EEP (Early Exercise Premium) component of greeks + the european greeks with respect to spot/vol/time. The issue with this method is there is no discontinuity of greeks I have derived beyond the boundary as in this example. What is the best way of approaching this? Thanks.