On continuous images of self-similar sets

Let (M,ck,nk,κ) be a class of homogeneous Moran sets. Suppose f(x,y)∈C3 is a function defined on R2. Given E1,E2∈(M,ck,nk,κ), in this paper, we prove, under some checkable conditions on the partial derivatives of f(x,y), thatf(E1,E2)={f(x,y):x∈E1,y∈E2} is exactly a closed interval or a union of finitely many closed intervals. Similar results for the homogeneous self-similar sets with arbitrary overlaps can be obtained. Further generalization is available for some inhomogeneous self-similar sets if we utilize the approximation theorem.