Functions characterized by images of sets

For non-empty topological spaces X and Y and arbitrary families ${\displaystyle calA}$ ⊆ ${\displaystyle calP\left(X\right)}$ and ${\displaystyle calB\subseteq calP\left(Y\right)}$ we put ${\displaystyle cal{C}_{calA,calB}}$={f ∈ ${\displaystyle Y^X}$ : (∀ A ∈ ${\displaystyle calA}$)(f[A] ∈ ${\displaystyle calB)}$}. We examine which classes of functions ${\displaystyle calF}$ ⊆ ${\displaystyle Y^X}$ can be represented as ${\displaystyle cal{C}_{calA,calB}}$. We are mainly interested in the case when ${\displaystyle calF=calC(X,Y)}$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class ${\displaystyle calF=calC}$(X,ℝ) is not equal to ${\displaystyle cal{C}_{calA,calB}}$ for any ${\displaystyle calA}$ ⊆ ${\displaystyle calP\left(X\right)}$ and ${\displaystyle calB}$ ⊆ ${\displaystyle calP}$(ℝ). Thus, ${\displaystyle calC}$(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 ${\displaystyle cal{C}_{calA,calB}}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.