Continuity of critical points for 1-dimensional non-local energies

Key Points

• The aim is to establish the continuity of bounded critical points for Riesz energy in one dimension under specific growth conditions.
• Analyzed the properties of Riesz energy with attractive-repulsive characteristics.
• Applied mathematical frameworks to assess kernel growth behavior at the origin.
• Utilized theoretical proofs to establish continuity within the support of critical points.
• Proved that bounded critical points of Riesz energy are continuous in their support.
• Demonstrated that growth conditions on the kernel significantly influence continuity.