subsets of R^2, continuity

hello all,

suppose that E, F are two connected subsets of R^2. Is there any
conditions on E,F that ensures that there exists a continuous function
f:E \to R^2 such that f(E)=F. Of course Card(E) >= Card(F), but it
doesn t seems sufficient. And if moreover one wants that f is an
injection ?

thank you,
Fedor

.