The main idea of this thesis is to demonstrate the connections of various semi-continuity properties to each other as well as to present that one can involve such results into game theory in a natural way. We investigate upper hemi-continuous set valued mappings in order to obtain selection and fixed point theorems. We apply Kakutani's fixed point theorem to the best response mapping of a concave game in order to obtain a general existence theorem in game theory. Specific pairs of interval valued mappings with semi-continuous limits are investigated as well.