DISCONTINUOUS NEURAL SELECTORS IN ORLICZSPACES: MODULAR CHEBYSHEV LIMITATIONS BEYOND \( Lᵖ \)

It is well known that -spaces p1 ă ă 8q exhibit a fundamental limitation:there exists no continuous map : Ñ , where is any finite union of finite-dimensional neural network spans, such that } ´p q} distp, q ` for all P and for a fixed ą 0. In this work, we extend this discontinuity phenomenon from strictly convex normed spaces to classical Orlicz spaces Φ pq over compact domains, where Φ is an -function satisfying the Δ 2 -condition and strict modular convexity.More precisely, we prove that no continuous neural selector : Φ pq Ñ can remain within of the modular best approximation generated by ridge-function spans.Our argument adapts the geometric theory of Chebyshev sets to the modular framework associated with the Orlicz modular Φ , thereby revealing intrinsic optimization discontinuities in neural approximation theory beyond the classical -setting.