Continuity at a Point

Topology · Axiom Academy

Understanding the local neighborhood characterization of continuous functions Let be a function between topological spaces and , and let . The function is continuous at if for every neighborhood of , the preimage is a neighborhood of . In other words, is continuous at if "nearby" points in map to "nearby" points in . The neighborhood condition formalizes this intuitive idea of "closeness" in the topological setting. Step 2: Open Set Characterization We can reformulate the definition using open neighborhoods: is continuous at if and only if for every open set containing , there exists an open set containing such that . This says: if we choose any open set around the output , we can find an open set around the input that maps entirely into our chosen output set. Step 3: From Local to Global Continuity A function is continuous (in the global sense) if and only if is continuous at every point . This connects our two notions of continuity: Global continuity: Preimages of open sets are open (everywhere at once) Local continuity: Preimages of neighborhoods are neighborhoods (at each point) The theorem tells us these are equivalent: being continuous everywhere means being continuous at each point, and vice versa. Step 4: Example - Continuous at a Point This is a neighborhood of , so is continuous at Step 5: Example - Discontinuous at a Point This function is not continuous at : But is not a neighborhood of ! Every neighborhood of contains irrational numbers

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