Limits of sequences of feebly-type continuous functions (Q2024125)

Let us introduce the following families of real-valued two variable functions: feebly continuous (FC), very feebly continuous (VFC), two-feebly continuous (TFC). The following are the main results of this exposition. Theorem 1. Every function \(f:\mathbb{R}^2\to \mathbb{R}\) is the pointwise limit of sequences \((f_n)\), \((g_n)\), \((h_n)\) in FC, VFC\(\setminus\)FC and TFC\(\setminus\)VFC respectively. Moreover, if \(f\) is Borel (Lebesgue, Baire) measurable, then all sequences \((f_n)\), \((g_n)\), \((h_n)\) have the same property. Consequently, FC, VFC\(\setminus\)FC and TFC\(\setminus\)VFC are dense in the space \(\mathbb{R}^{\mathbb{R}^2}\) with the pointwise convergence topology. Theorem 2. The families VFC and TFC are closed in the topology of uniform convergence in \(\mathbb{R}^{\mathbb{R}^2}\). Theorem 3. The family TFC\(\setminus\)VFC is uniformly dense in TFC. Consequently, VFC is nowhere dense in TFC with the topology of uniform convergence. Theorem 4. The family FC is not closed in the space \(\mathbb{R}^{\mathbb{R}^2}\) with the uniform convergence topology. Theorem 5. The family FC is not uniformly dense in the class VFC. Theorem 6. The family VFC\(\setminus\)FC is uniformly dense in VFC. Further aspects occasioned by these developments are also discussed.