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.**