Proving Continuity

Topology · Axiom Academy

Three worked examples with step-by-step proofs Problem 1: Continuity of f(x) = x² Prove that is continuous from using the open set definition. Why is the preimage of an open set U under f(x) = x² itself open? Problem 2: Continuity of Projection Prove that the projection map defined by is continuous. What makes the projection map continuous? Problem 3: Floor Function Discontinuity Show that the floor function from is NOT continuous. What specific property fails for the floor function? You've completed all three continuity proofs: 1. Proved f(x) = x² is continuous using open sets 2. Proved the projection map is continuous 3. Showed the floor function is NOT continuous These examples demonstrate the power of the open set definition of continuity.

This is the written version of the interactive lesson above. See the full Topology course.