Pareto efficient allocation and core allocation in 2 person exchange economy with discontinuous utility
Here are two pertinent observations:


If A has a positive amount of the $x$-good, and $B$ a positive amount of the $y$-good, both can be made strictly better off (why?).
There is no efficient allocation in which $A$ has no amount of the $x$-good, but a positive amount of the $y$-good (why?).


Remaining are the allocations in which $B$ gets everything and allocations in which A has a positive amount of the $x$-good and all of the $y$-good. Verify that these are indeed Pareto efficient allocations. 

Finding core allocations is not much different. There is a small difference if you define blocking so that all parties involved might be strictly better off. In that case, you can have allocations in which $A$ has none of the $x$-good but some of the $y$-good.