A Polish group has the automatic continuity property if every homomorphism to a separable group is continuous. Some mapping class groups of infinite-type surfaces are known to have this property, and some are known not to. We will discuss how to build discontinuous homomorphisms given certain nice families of nondisplaceable subsurfaces. We then will see how we can loosen this nondisplaceable condition to see the failure of automatic continuity for pure mapping class groups. Part of this work is joint with Ryan Dickmann and part is in progress with Mladen Bestvina and Kasra Rafi.