{"id":"continuity","domain":"analysis","type":"Concept","title":"Continuity","slug":"continuity","url":"/mathematics/analysis/continuity/","summary":"A function with no sudden jumps or breaks. Defined rigorously via limits. Underlies Intermediate Value Theorem and Extreme Value Theorem.","formal_definition":"f continuous at a: lim(x->a) f(x) = f(a)","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed."},{"tier":1,"citation":"Munkres, J.R. (2000). Topology. 2nd ed."},{"tier":2,"citation":"Spivak, M. (2008). Calculus. 4th ed."},{"tier":3,"citation":"Bressoud, D.M. (2007). A Radical Approach to Real Analysis. 2nd ed."}],"relationships":[{"type":"built_from","target":"limit","label":"Limit"},{"type":"required_for","target":"derivative","label":"Derivative"},{"type":"generalised_by","target":"topological-space","label":"Topological Space"},{"type":"used_in","target":"differential-equation","label":"Differential Equation — Lipschitz continuity"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.1"}