Closed graph theorem

The Closed Graph Theorem states that if a linear function between certain mathematical spaces (called Banach spaces) has a graph that is "closed"—meaning it contains all its limit points—then the function is necessarily continuous or "well-behaved." In practical terms, this theorem assures us that whenever the function's behavior is predictable in terms of limits, we can confidently say it works smoothly without surprises. It provides a fundamental link between a function’s geometric properties (its graph) and its analytical properties (continuity), ensuring stability and consistency in mathematical analysis.