Continuity Beyond the Real Numbers

An expository article on using continuity for objects other than real numbers (matrices, pairs of matrices, functions, lines and geometric configurations), written with high-school tools only. It starts from three Romanian olympiad problems (National Olympiad 2022, county rounds 2026 and 2004) whose solutions share the same structure: the property is proved on a convenient class of objects, then extended to all objects by approximation. From them it builds a general framework (objects described by parameters, the largest-deviation distance, continuous quantities, dense classes and an extension principle, continuous families and the Darboux property) and applies it to IMC 2004, a problem from Gazeta Matematică, the Romanian National Olympiad 2009 and a new variant of it. A separate section shows when the argument fails (a dense class outside the class of the hypothesis, rank and similarity, the length of a graph). The exercises include Putnam 2003 A2, IMC 2004, RMM Shortlist 2017 G1 and a problem from the Tournament of Towns. Available in English, Romanian and French. Author website: https://vladconstantinescu.com