In this paper necessary and
sufficient conditions are developed for certain classes of continuous functions
f(X) = Y , where X and Y are arbitrary metric spaces, to have the property that
dimK =dimf(K) for all closed K ⊂ X. In particular it is shown that if f is closed
and dimf(K) >dimK for some closed K ⊂ X, then there exists a closed K′⊂ X so
that dimK′ = 0 and dimf(K′) > 0. These results are then used to show that if f is
closed and finite to one so that the multiplicity function of f takes on at most
k + 1 distinct values, then dimK ≦dimf(K) ≦dimK + k for all closed
K ⊂ X.
