The Modified Newtonian Dynamics (MOND) paradigm generically predicts that the external gravitational field in which a system is embedded can produce effects on its internal dynamics. In this communication, we first show that this external field effect (EFE) can significantly improve some galactic rotation curves fits by decreasing the predicted velocities of the external part of the rotation curves. In modified gravity versions of MOND, this EFE also appears in the Solar system and leads to a very good way to constrain the transition function of the theory. A combined analysis of the galactic rotation curves and Solar system constraints (provided by the Cassini spacecraft) rules out several classes of popular MOND transition functions, but leaves others viable. Moreover, we show that Laser Interferometer Space Antenna Pathfinder will not be able to improve the current constraints on these still viable transition functions.