On a problem related to a non-squeezing theorem

In this paper we give a solution to a problem of Kulpa about the interior of the image of certain continuous maps f : X → R n where X is a compact subset of R n with non-empty interior. Moreover we show that the image of every continuous map f : X → R 2 where X is a non-empty compact subset of R 2 and diam f −1 f ( x ) < 3 a X for every x ∈ Fr X , has non-empty interior.