[Paper Review] Points of continuity of quasiconvex functions on topological vector spaces

Patrick J. Rabier|arXiv (Cornell University)|Jun 25, 2012

Optimization and Variational Analysis6 references3 citations

TL;DR

This paper establishes necessary and sufficient topological conditions on the lower level sets of a real-valued quasiconvex function on a Baire topological vector space for the function to be continuous on a residual (dense Gδ) subset. The key result shows that discontinuities are of first category if and only if each sublevel set $ F_\alpha = \{x : f(x) < \alpha\} $ is nowhere dense whenever its closure has empty interior, leveraging a novel concept of topological essential extrema to analyze continuity structure.

ABSTRACT

We give necessary and sufficient conditions for a real-valued quasiconvex function f on a Baire topological vector space X (in particular, Banach or Frechet space) to be continuous at the points of a residual subset of X. These conditions involve only simple topological properties of the lower level sets of f. A main ingredient consists in taking advantage of a remarkable property of quasiconvex functions relative to a topological variant of essential extrema on the open subsets of X. One application is that if f is quasiconvex and continuous at the points of a residual subset of X, then with a single possible exception, f^{-1}(a) is nowhere dense or has nonempty interior, as is the case for everywhere continuous functions. As a barely off-key complement, we also prove that every usc quasiconvex function is quasicontinuous in the (classical) sense of Kempisty since this interesting property does not seem to have been noticed before.

Motivation & Objective

• To characterize the set of continuity points of real-valued quasiconvex functions on Baire topological vector spaces.
• To identify simple topological conditions on sublevel sets that ensure continuity on a residual subset.
• To establish a category-theoretic analogue of Crouzeix’s differentiability result in finite dimensions.
• To explore the relationship between quasiconvexity and quasicontinuity, particularly for upper semicontinuous functions.
• To demonstrate that the set of discontinuities of such functions is always of first category under mild topological conditions on sublevel sets.

Proposed method

• Introduces the concept of topological essential extrema $ \mathcal{T}\text{ess}\sup_U f $ and $ \mathcal{T}\text{ess}\inf_U f $, defined via first category sets, to analyze extremal behavior in open subsets.
• Uses the Baire category theorem in the context of topological vector spaces to analyze the structure of sublevel sets $ F_\alpha $ and $ F_\alpha' $.
• Applies the notion of residual sets and first category sets to characterize the continuity points of quasiconvex functions.
• Establishes that a quasiconvex function is continuous on a residual set if and only if $ F_\alpha $ is nowhere dense whenever $ \overset{\circ}{F_\alpha'} = \emptyset $, linking topological properties of sublevel sets to continuity.
• Proves that upper semicontinuous quasiconvex functions are quasicontinuous in the sense of Kempisty by showing the interior of $ f^{-1}(\Omega) $ is dense in $ f^{-1}(\Omega) $ for open $ \Omega \subset \mathbb{R} $.
• Utilizes the fact that $ \mathcal{T}\text{ess}\inf_X f $ is invariant under modification on first category sets, enabling robust analysis of extremal behavior.

Experimental results

Research questions

• RQ1 Under what topological conditions on sublevel sets is a quasiconvex function continuous on a residual subset of a Baire topological vector space?
• RQ2 How does the structure of the sublevel sets $ F_\alpha $ and $ F_\alpha' $ relate to the category of the set of discontinuities?
• RQ3 Can the concept of topological essential extrema be used to derive continuity criteria for quasiconvex functions?
• RQ4 Is every upper semicontinuous quasiconvex function quasicontinuous in the sense of Kempisty?
• RQ5 To what extent do the continuity properties of quasiconvex functions in infinite-dimensional spaces resemble those in finite-dimensional spaces?

Key findings

• A real-valued quasiconvex function on a Baire topological vector space is continuous on a residual subset if and only if, for every $ \alpha \in \mathbb{R} $, the set $ F_\alpha = \{x : f(x) < \alpha\} $ is nowhere dense whenever the interior of $ F_\alpha' = \{x : f(x) \leq \alpha\} $ is empty.
• The set of discontinuities of a quasiconvex function on a Baire space is always of first category under the stated condition on sublevel sets.
• Every upper semicontinuous quasiconvex function on a topological vector space is quasicontinuous in the sense of Kempisty, meaning the interior of $ f^{-1}(\Omega) $ is dense in $ f^{-1}(\Omega) $ for every open $ \Omega \subset \mathbb{R} $.
• The result provides a category-theoretic analogue of Crouzeix’s theorem: quasiconvex functions on Baire spaces are continuous on a residual set, even without differentiability.
• In non-metrizable or non-locally convex spaces like $ L^p(0,1) $ for $ 0 < p < 1 $, the condition implies that quasiconvex functions continuous on a residual set must be constant on a residual set.
• The concept of topological essential extrema is shown to be invariant under modification on first category sets and is instrumental in characterizing continuity structure.