For non-empty topological spaces X and Y and arbitrary families A 'subset or equal' P(X) and B 'subset or equal' P(Y) we put C _{A, B} = {f \in Y^X : ( 'for all'A \in A)(f[A] \in B)}. In this paper we will examine which classes of functions F 'subset or equal' Y^X can be represented as C _{A, B}. We will be mainly interested in the case when F = C(X,Y) is the class of all continuous functions from X into Y. We prove that for non-discrete Tychonoff space X the class F = C(X,R) is not equal to C _{A, B} for any A 'subset or equal' P(X) and B 'subset or equal' P(R). Thus, C(X,R) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as C _{A, B}: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.

Mathematics Subject Classification: 54C05 (26A15, 18B30, 54C30)
Keywords: Tychonoff space, functionally Hausdorff space, Cook continuum, strongly rigid family of spaces, continuous function, upper or lower semicontinuous function, derivative, approximately continuous function, Baire class 1 function, Borel function, measurable function

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Date received: September 22, 1996.
Date published: September 30, 1996.