  Let $(\mathcal{M}, c_k, n_k,\kappa)$ be a class of homogeneous Moran sets.
Suppose $f(x,y)\in C^3$ is a function defined on $\mathbb{R}^2$. Given $E_1,
E_2\in(\mathcal{M}, c_k, n_k,\kappa) $, in this paper, we prove, under some
checkable conditions on the partial derivatives of $f(x,y)$, that
$$f(E_1,E_2)=\{f(x,y):x\in E_1,y\in E_2\}$$ 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.
