For arbitrary families A and B of subsets of R let

C(A,B)= {f| f: R-->R and the image f[A] is in B for every A in A}

and

$C-1$ (A,B)= {f| f: R-->R and the inverse image $f-1$(B) is in A for every B in B}.

A family F of real functions is characterizable by images (preimages) of sets if F=C(A,B) (F=$C-1$(A,B), respectively) for some families A and B. We study which of classes of Darboux like functions can be characterized in this way. Moreover, we prove that the class of all Sierpinski-Zygmund functions can be characterized by neither images nor preimages of sets.

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Last modified September 10, 1998.