In this note we establish two independent results on fixed points. The first one is about the continuity of fixed points of limit-compact mappings introduced by Sadovskii in [7]. This result partially generalizes a result of Tan in [10] for collectively condensing mappings. The second result is a new fixed point theorem for the sum of a generalized contraction and a compact mapping, which improves a well-known result of Krasnoselskii [4]. A probabilistic version of this result is also presented here.