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For non-empty topological spaces X and Y and arbitrary families $𝒜$ ⊆ $𝒫\left(X\right)$ and $ℬ\subseteq 𝒫\left(Y\right)$ we put ${𝒞}_{𝒜,ℬ}$=f ∈ $Y^X$ : (∀ A ∈ $𝒜$)(f[A] ∈ $ℬ)$. We examine which classes of functions $ℱ$ ⊆ $Y^X$ can be represented as ${𝒞}_{𝒜,ℬ}$. We are 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 a non-discrete Tikhonov space X the class $ℱ=𝒞$(X,ℝ) is not equal to ${𝒞}_{𝒜,ℬ}$ for any $𝒜$ ⊆ $𝒫\left(X\right)$ and $ℬ$ ⊆ $𝒫$(ℝ). Thus, $𝒞$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...